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 will power. 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: The Possession Experiment.
Guild of Biomancers
Project: INFERNAL KETER
RESEARCH LOG NO. 12.789.15
We have developed a scientific hypothesis on how metaphysical entities manage to possess and interact with sentient intelligent lifeforms. Currently we and the Guild of Technomancers are setting up a series of experiments to test this out.
We will be first conducting a series of observational studies at various temples to monitor the brain activity of worshippers with fMRI arrays as they pray to and perform the rituals to gain their deity's favor. These will be known as OBSERVATION TEST: I-KETER 01 - 10