Bodyswap info
Physical
His appetite and need for sleep are a major thing. Someone used to sleeping eight or nine hours a night will find themselves craving naps the whole time they're awake, and capable of sleeping much longer at a stretch. If they're used to eating three square meals a day, they'll suddenly be starving much of the rest of the time. Axel's body always needs more fuel, whether it's rest or food or both.
Chakra is different from any other type of energy that one can draw on in the dressing room, whether it's light or darkness or simply mana for spells. It's more balanced, harder to notice if one isn't used to it, and it can be quite difficult to grab hold of and use. This is complicated by Axel's two kinds of chakra, and how very powerful Hachibi's is. But there's no danger of spontaneous fire or anything, as there's a specific process of using the chakra to produce fire.
Axel's senses are absurd compared to a normal person's, especially his sense of smell. It's enough information flooding in that a stranger would be overwhelmed in a hurry, if they could make sense of it. But scent is a language unto itself, and if you're not familiar with it, then it's all just gibberish. Axel can tell one scent from another through experience, and trial and error. For anyone else, the impression would just be SMELL, rather than all of the information about eating/sex/other human contact/bathroom habits/etc. that Axel gets from a single person.
His hearing and sight are also quite sharp, but only as much as any ninja, which is only a bit better than a normal human's.
Mental
Hachibi's fucking insane and sadistic, let's start with that. He's pretty much settled and resigned to having Axel as his host, and in fact Axel is the only host he's actually liked. But. This is mostly due to Axel's personality and mindset, which includes the fact that he has never viewed Hachibi as a source of power that he can control. He knows Hachibi could easily rip him apart from the inside. And he almost views Hachibi as a kind of sensei, and definitely as a friend or even brother. Hachibi reciprocates a lot of these feelings, but someone else in there besides Axel would completely negate that.
There's also the matter of Hachibi's sheer volume. He doesn't have an audible voice (to his host) most of the time, only when he really needs to get Axel's attention for some reason. He's more of a loud presence, loud thoughts, vivid images. Axel's used to it, and knows how tune the unimportant parts out, but it took him years to get that way. One of the best examples of this is with tinyxel, who is still fairly new to the whole jinchuuriki thing, and is prone to interrupting himself to respond to Hachibi out loud, and responding physically with flinching and the like if the conversation is unpleasant, which it often is. The physical reactions are due to the "loudness" and the inherent violence behind the words a lot of the time. The general impression a stranger to all this would have in their first encounters with Hachibi would be one of lots of mental static, bigger shocks similar to sudden thunder or something very large being dropped right next to them, "hearing" a voice that is deep and animalistic, and a feeling of something huge being right up in their personal space. Basically, to the untrained, even a simple greeting from Hachibi would be like having Lexaeus standing at your side, speaking rather loudly directly into your ear, and hitting you across the back with every word. But all in your mind.
When Axel meditates and goes to commune with Hachibi on more even ground, or switch places with him, where he goes is purely a visualization. It's easier for everyone that way, rather than trying to wade through the technical aspects of "okay, this is my chakra and this other chakra is Hachibi's, and I am talking to a disembodied voice that lives in my body". It's much better with visuals, and being able to mentally visit an area that Hachibi is caged inside of, and touch him, pull his ears, climb on him, all of that. That way, when they switch places, Axel has a place he can stay grounded in, mentally (if he's even awake for it). The same visualization is more of a reality for Hachibi, in that he really is just chakra with a voice and an intricate connection to an actual living (human) body. He helps create it as a mental image with Axel, and maintains it most of the time, because really, what else does he have to do? It's also a convenient way to have access to whatever he needs of Axel's; the caged area is a vast expanse of sandy wasteland, with nearly every grain of sand representing a thought or a memory or a minute bodily status change. On a more technical level, he's tapped directly into Axel's body and consciousness through their shared chakra pathways, aware of things like body temperature, chakra levels for both of them, hunger, energy, all of the senses (except taste, which is largely ignored, mostly because it makes him grumpy that he's not actually capable of eating anymore), dreams, passing thoughts, right down to muscle twitches and . But just as Axel filters out Hachibi's unimportant business and doings, Hachibi largely ignores much of this information.
Hachibi's chakra changes based on his mood, and it generally manifests as an increase in energy (if it's a positive mood) or a physical sensation like nausea (if it's a negative mood, such as anger or resentment). There's also a subtle shift of consciousness between when he's actively accessing Axel's senses to look out at the world with him, and when he's not. It's impossible to describe, but there's definitely a different feeling involved.
Kai
Kai will most likely have a different experience than anyone else would, because it's Kai. Normally, Hachibi is calm and moderately well-behaved around him, because of the whole pure heart and Light thing. In the event of a body swap between Kai and Axel, the effects of Kai's body will still be there, and he has no reason to try and mess with Kai personally while they're sharing headspace. Anyone else would have him raging at them, trying to actively take control and drive them insane enough that they won't fight back, all the things he usually does when forced into a new host. BUT, with Kai, he's also not going to make any effort to tone things down, do anything differently, etc. That's just how he is. He'll still be loud, he'll still butt into thoughts and possibly take over (which would dump Kai into the mental prison, even if Kai wasn't visualizing such a thing, which he probably wouldn't be), and he'll still be rude and inclined to violent thoughts.
References
Point 10 of the 10 things meme, here.
Prose, here. The first dream is purely a memory, but describes how Hachibi reacts to a new host body. The other dreams are corruptions, and are a good example of the kind of imagery Hachibi's fond of creating for lulz.
His appetite and need for sleep are a major thing. Someone used to sleeping eight or nine hours a night will find themselves craving naps the whole time they're awake, and capable of sleeping much longer at a stretch. If they're used to eating three square meals a day, they'll suddenly be starving much of the rest of the time. Axel's body always needs more fuel, whether it's rest or food or both.
Chakra is different from any other type of energy that one can draw on in the dressing room, whether it's light or darkness or simply mana for spells. It's more balanced, harder to notice if one isn't used to it, and it can be quite difficult to grab hold of and use. This is complicated by Axel's two kinds of chakra, and how very powerful Hachibi's is. But there's no danger of spontaneous fire or anything, as there's a specific process of using the chakra to produce fire.
Axel's senses are absurd compared to a normal person's, especially his sense of smell. It's enough information flooding in that a stranger would be overwhelmed in a hurry, if they could make sense of it. But scent is a language unto itself, and if you're not familiar with it, then it's all just gibberish. Axel can tell one scent from another through experience, and trial and error. For anyone else, the impression would just be SMELL, rather than all of the information about eating/sex/other human contact/bathroom habits/etc. that Axel gets from a single person.
His hearing and sight are also quite sharp, but only as much as any ninja, which is only a bit better than a normal human's.
Mental
Hachibi's fucking insane and sadistic, let's start with that. He's pretty much settled and resigned to having Axel as his host, and in fact Axel is the only host he's actually liked. But. This is mostly due to Axel's personality and mindset, which includes the fact that he has never viewed Hachibi as a source of power that he can control. He knows Hachibi could easily rip him apart from the inside. And he almost views Hachibi as a kind of sensei, and definitely as a friend or even brother. Hachibi reciprocates a lot of these feelings, but someone else in there besides Axel would completely negate that.
There's also the matter of Hachibi's sheer volume. He doesn't have an audible voice (to his host) most of the time, only when he really needs to get Axel's attention for some reason. He's more of a loud presence, loud thoughts, vivid images. Axel's used to it, and knows how tune the unimportant parts out, but it took him years to get that way. One of the best examples of this is with tinyxel, who is still fairly new to the whole jinchuuriki thing, and is prone to interrupting himself to respond to Hachibi out loud, and responding physically with flinching and the like if the conversation is unpleasant, which it often is. The physical reactions are due to the "loudness" and the inherent violence behind the words a lot of the time. The general impression a stranger to all this would have in their first encounters with Hachibi would be one of lots of mental static, bigger shocks similar to sudden thunder or something very large being dropped right next to them, "hearing" a voice that is deep and animalistic, and a feeling of something huge being right up in their personal space. Basically, to the untrained, even a simple greeting from Hachibi would be like having Lexaeus standing at your side, speaking rather loudly directly into your ear, and hitting you across the back with every word. But all in your mind.
When Axel meditates and goes to commune with Hachibi on more even ground, or switch places with him, where he goes is purely a visualization. It's easier for everyone that way, rather than trying to wade through the technical aspects of "okay, this is my chakra and this other chakra is Hachibi's, and I am talking to a disembodied voice that lives in my body". It's much better with visuals, and being able to mentally visit an area that Hachibi is caged inside of, and touch him, pull his ears, climb on him, all of that. That way, when they switch places, Axel has a place he can stay grounded in, mentally (if he's even awake for it). The same visualization is more of a reality for Hachibi, in that he really is just chakra with a voice and an intricate connection to an actual living (human) body. He helps create it as a mental image with Axel, and maintains it most of the time, because really, what else does he have to do? It's also a convenient way to have access to whatever he needs of Axel's; the caged area is a vast expanse of sandy wasteland, with nearly every grain of sand representing a thought or a memory or a minute bodily status change. On a more technical level, he's tapped directly into Axel's body and consciousness through their shared chakra pathways, aware of things like body temperature, chakra levels for both of them, hunger, energy, all of the senses (except taste, which is largely ignored, mostly because it makes him grumpy that he's not actually capable of eating anymore), dreams, passing thoughts, right down to muscle twitches and . But just as Axel filters out Hachibi's unimportant business and doings, Hachibi largely ignores much of this information.
Hachibi's chakra changes based on his mood, and it generally manifests as an increase in energy (if it's a positive mood) or a physical sensation like nausea (if it's a negative mood, such as anger or resentment). There's also a subtle shift of consciousness between when he's actively accessing Axel's senses to look out at the world with him, and when he's not. It's impossible to describe, but there's definitely a different feeling involved.
Kai
Kai will most likely have a different experience than anyone else would, because it's Kai. Normally, Hachibi is calm and moderately well-behaved around him, because of the whole pure heart and Light thing. In the event of a body swap between Kai and Axel, the effects of Kai's body will still be there, and he has no reason to try and mess with Kai personally while they're sharing headspace. Anyone else would have him raging at them, trying to actively take control and drive them insane enough that they won't fight back, all the things he usually does when forced into a new host. BUT, with Kai, he's also not going to make any effort to tone things down, do anything differently, etc. That's just how he is. He'll still be loud, he'll still butt into thoughts and possibly take over (which would dump Kai into the mental prison, even if Kai wasn't visualizing such a thing, which he probably wouldn't be), and he'll still be rude and inclined to violent thoughts.
References
Point 10 of the 10 things meme, here.
Prose, here. The first dream is purely a memory, but describes how Hachibi reacts to a new host body. The other dreams are corruptions, and are a good example of the kind of imagery Hachibi's fond of creating for lulz.