[jmccree]

I'm not sure if you're referring to a certain jurisdiction in the US, another country, or speaking "theoretically per your interpretation of the US constitution if a case made it to the supreme court", but that is not accurate for the majority of the states in the US. Most US states I'm aware of currently grant officers police authority on and off duty per state law.

[e12e]

Hm, I see that police in Norway also seem to have the same authority off-duty as while at work (and certain obligation to intervene if they happen upon serious crime, as well as the option to intervene as if they where on duty (as long as they are able to identify themselves as police officers, and are sober)). They are prohibited against taking other jobs (moonlighting) without approval from the police commissioner, however.