There is a lot of "what if" there. But it would depend on the officer so I can't give you like a be all end all answer. But I could tell you what I would do.
10-4, I would suspect that a lot of cops would use their "I'm a cop" saying to try to gain instant control of the situation. That is where I believe they crossed that fine line. Either they are a security officer or they are a cop. (see my point?) You are right, it does depend on who the person is.
I feel sure that the businesses that use the off duty cops feel more secure hiring an off duty cop as opposed to a mall cop. Also, I feel sure it is a good way for off duty cops to make some extra cash. It's a win win situation for most.