This article contains fan theories about Sister Mary Eunice's possession and the nature of demons. It is separate to avoid contaminating the main page with spoilers and speculation. It is subjective.
Sister Mary Eunice is not totally possessed, the Devil only awaked her evil side. And he is not in total control, on the contrary, she is controling him.
Sister Mary Eunice is not totally possessed
# Jed's eyes were yellow all the time (meaning a total possession) and Sister Mary Eunice's turn in yellow only when she loses control.
Sister Devil Eunice don't act like a possessed person, she is not spinning her head, cursing or eating hearts. She is just acting evil and manipulating people.
When she was talking with Jenny in "The Origins Of Monstrosity", she said that she was the devil, but she acted like she changed by own herself, not by the demon inside her.
When dancing to "You Don't Own Me" she appeared to be revolting against God, and choosing Satan. This proves that she is changing her way to think, her personality, but not her spirit.
When talking to Shachath in "Dark Cousin", Sister Mary Eunice starts crying and says "Oh heavenly host, will you release me? Can you release me?" until the devil says "Oh shut up you stupid sow!" Then the devil take control and says "She likes it here, we like it.". This could mean that she likes the power the the devil gives to her, but she want him to leave.
When talking to Shachath(on "Dark Cousin") she says "Oh heavenly host, will you release me? Can you release me?". Like most people that met the angel, she could be under Shachath's influence(most people feel safe, and relaxed) not acting on her own will.
When Mary Eunice pleads to Shachath, and the Devil takes over again, there is an instant personality change like the body is totally taken over.
When Sister Mary Eunice speaks to the Monsignor in "The Name Game", she briefly regains control of her body and weeps, saying that she is "tired of fighting" and she wants to "let go" of her life. This implies that she was trying to resist all along against the demon that was possessing her but to no avail.
Sister Mary Eunice might be can't be completely possessess by The Demon because she is a nun.