The answer to that question is: precisely. Exorcists get a lot of horror mileage, but what about the victims? There are different ways that the victim can be handled for varying amount of horror. Most times the possessed person is not aware of what happens when the possessing entity is in control of their body. If the focus is on the victim there is a fair bit of play in that as they learn about the heinous things they were doing. Often the victim might learn about the events that have transpired at the same time as the audience. Other times the suspense and anticipation of seeing the victim?s response to the news of their unconscious and uncontrolled actions that the audience is fully aware of is desirable. This works as soon as the audience sees what the possessed is doing. It can be heightened by near misses with the truth after and with the complications of those actions interacting with other important characters in the story.
The next option works best, if not only, in written fiction. The possessed at the author's discretion is fully aware of what they are doing while the possessing entity is in control of their body. Or at least that's what they believe. This is where some of the best horror comes from. Trapped in your own mind watching while your body does unspeakable things and says terrible things to anyone and everyone damaging your credibility, threatening your freedom, and even putting you in danger both while still possessed and potentially for a long time after the possession is over. The entity might give up control and let you deal with everything it has done only to take control again once everything has been straightened out and begin the entire nightmare process over and over again. It would also make certain that you knew it hid some events from you just to add to your already tremendous sense of worry.
What if the possessed is doing things of their own will, but with the dark influence of the entity egging them on, acting as or suppressing their conscience? The influence could begin long before the first act of obvious possession. It could begin years in advance. This could be par for the course and full on possession takes place when the corruption doesn't happen fast enough or the victim is too resist. Or it could be part of an even more nefarious plot. These are the intimate horrors associated with possession. What happens though when a larger look at the phenomena is taken? Possessed people might be secretly meeting and planning much larger events than any single person might be able to pull off even with supernatural abilities at their beck and call. If we are talking about purely Satanic-based possession then the overall plan is far reaching across not only distance but also time, and the end game is frighteningly unknown. Hints about that end game may be worse than knowing nothing at all.
Music: Hell No by Bruce Dickinson and Hitchin' A Ride by Green Day.