It depends on how possessed the person was/is,how strong or powerful the demon operates at that point in time.
In most cases the possessed isn't aware of what was happening.
Except the person(possessed) has mastered the demon,gotten used to it or aware that he or she is indeed possessed with such demon.
In some cases, if the person flesh is strong and the demon isn't strong enough to take over the whole body the possessed could be aware of what goes on within his/her body but may not be totally in control
To answer your question, yes, it's possible the possessed is conscious of what was going on, and while I explained above that, in such cases the person's body has gotten used to the demon or mastered the demon.
So, Yes! It's possible
Bless u