I helped put my ex-BIL through the Police Academy.
IIRC once you are a sworn peace officer you are one 24x7x365. Even if you are "off duty" you still have to enforce the law. If you see a crime you have to attempt to stop it and apprehend the suspects. The difference is you don't go looking for crime when you are off duty.
And I also think they have to have their service weapons with or near them at all times.
In some departments that is true, but many cities are not this way: They will bust an off-duty cop for carrying as quickly as anybody else.
Plus, a cop in plain clothes has to be very careful about getting involved in police action whether it is department policy or not. Just a couple of months ago an off duty cop was shot and killed in Houston when he joined a foot chase armed. Another officer mistook him for a suspect.
Louisiana cops have even been known to bust Texas cops in uniform and on duty that cross the state line. Cops aren't super-citizens, and depending on how the law is written in a given locale there may not be a lot of difference between them and non-officers in terms of carrying and deadly force.