I would say like any other employer, but ESPECIALLY Disney, anyone who was caught blogging or posting on a board about their current or previous experiences as a character would lose their job on the spot. Disney is all about the magic. Can you imagine someone walking up to them in a store and yelling out, "Hey, there is the woman that is Minnie Mouse."
That being said, I work very closely with a guy whose wife grew up in Orlando and her closest childhood friend still lives there. Of course, I would never be meeting her and even if I did, she would never say it to me, but he confirmed she is one of the Minnie Mouses (yes, folks there is more than one).
She says that virtually most of the people that work AT Disney don't even know what she does - characters, especially Mickey and Minnie are very guarded characters that go through several layers of the building before getting into costume.