When an entity possesses or interacts with someone, it can manifest in two ways.
First is in the physical world for all to perceive. The second is inside you brain where only you can perceive it. But for this we will talk about the second one.
First things first, your brain is the ultimate controller of your body and can override your conscious willpower. What your brain decides, goes and you have no choice in its decisions.
When an entity possesses or interacts with you and you hear it speaking to you, or see it, or feel it, smell it etc. if no one else can perceive it, then you know it has taken up residence in your brain and the brains of anyone who it wants to be perceived by.
Basically think of your brain as a computer that controls the body and the entity as a hacker that is controlling it.
You see, your brain takes in stimuli from the sensory organs such as the eyes, nose, ears, skin, tongue etc. That stimuli is then processed by your brain. So when an entity makes a limited number of people perceive it, it is feeding the appropriate stimuli directly into the brain bypassing the sensory organs.
So if it wants to be heard, it will input auditory stimuli. If it wants to be seen, it will input visual stimuli. You get the idea here. This is why it looks so real to its victims but not to anyone else. It is real but by sending stimuli directly into the brain and bypassing the sensory organs, it does not need to manifest itself in the physical world for all to perceive.
Now if it is making people perceive things wrong such as allies into enemies etc, it is because the entity is intercepting the stimuli and modifying it before allowing the brain to process it. It can override conscious thoughts with its own making the body respond accordingly against the person's will by manipulating verious areas of the brain. It can block stimuli from reaching the brain thus making the person unable to see, hear, feel etc.
It can load experiences from a person's memory causing flashbacks, visions etc. It can edit memories. It can also generate its own dreams while the person sleeps. It can also override a person's personality with its own to completely take over a body.
As a result people can mistake these effects as some form of mental illness.
You can use these explanations if your players demand a more scientific justification for such situations.
A short story for some fluff:
Holy Temple of REDACTED
Interview with Sir Darken Rahl conducted by our priest Aralen.
Aralen: So Sir Rahl, How is it that you managed to exorcise the demon from your mind and body?
Darken Rahl: Oh, I didn't banish Amozabael from myself. He's still with me. I just found a way to allow my mind to ignore him for awhile.
Aralan: How so?
Darken Rahl: Well I discovered how Amoz was communicating with me with help from the Guild of Biomancers and figured out how to subvert it.
Darken Rahl: So the body takes in external stimuli from the senses of sight, sound, scent, touch, and taste. There are these organs in the body that convert these sensations to electrical signals which are sent to the brain for processing in to something that is recognizable. Well, Amoz can intercept these signals and modify them so that they feel like something else. In this way, the brain is tricked into sensing illusions without others knowing. He can also create these signals himself and feed them directly to the brain bypassing the organs completely. This is how the voices and visions seem so real.
Aralen: So this demon can create sensations that only you can perceive?
Darken Rahl: Exactly.
Aralen: Interesting. Now explain to me how you managed to stop these sensations.
Darken Rahl: Well, its kind of complex. It involves some extremely complicated alchemy.
Aralen: Try me.
Darken Rahl: Some beings on a distant planet known as Earth managed to discover an elixir that allows the brain to ignore Amozabael's manufactured sensations. In fact there are multiple formulas for this. I just picked the one that is the most effective. It has a name that is extremely hard to spell or pronounce. It is extremely complicated to brew and requires the creation of exotic substances mixed exactly as the formula specifies. It doesn't come in a potion though. Rather it comes in a solid about the size of a small diamond.
Aralen: Does it work for everyone?
Darken Rahl: In theory it should, but it can be dangerous for some people to take due to some side effects. In fact, it is not even known if it works on any of the other races. Only humans have been tested. On Earth they call it an "Anti-psychotic". So when Amoz will not stop bugging me, I just take one of these and he cannot communicate with me in any way for the next 3 hours or so unless I explicitly allow him to or summon him to this plane.
Aralen: So he is still with you?
Darken Rahl: Yes, and he will always be with me no matter what happens. This is not some magic demon banishing potion. It is a substance to help me ignore his attempts to interact with me from inside me.
Aralen: Well, that is all I needed to know. So thanks for telling me.
Darken Rahl: No problem.