Regarding Louisville, if memory serves, they are required to carry off-duty.
This is not Louisville, but my agency.
B. Officers are not required to be armed while OFF DUTY. If a firearm is carried off duty, under the umbrella of police authority (as opposed to as a citizen with a CCDW permit), the following guidelines will be followed:
1. The firearm shall be concealed at all times when in public, or when one could reasonably expect to encounter non-enforcement personnel, except at Headquarters or at home.----
2. The badge and police identification card shall be in the officer’s possession at all times, except at home.
3. Off duty officers who consume alcohol or plan to consume alcohol shall not be armed.
If I want to OC, I would have to put the badge away and make no claim of being an officer. At least that's my read on it. I find that it is interesting that the only alternative to carrying as an officer is carrying under concealed permit. Open carry is not really addressed.