Cops probably do get away with more things than the common citizen, but I know of one example where he didn't. A Morgan County off-duty deputy was shopping at a South Huntsville Wal Mart (Madison County) a month or so ago and saw a man who was wanted for something. Apparently he tried to arrest the guy (not sure he has jurisdiction in Madison County), but he wound up shooting the guy in the butt. He said it was accidental, but witnesses said he aimed and fired ... it was on purpose. He was charged with felony assault with a deadly weapon, as I recall.
But off duty cops definitely get away with things that we are "arrested" for, such as speeding. I've heard, but not sure of, that even family members of policemen can put a little blue stripe on their window that means "I'm part of the police family ... immune to traffic violations". I hope that's not true.