Cops can only operate within their own borders (like a NY cop cannot enforce the law in Canada) unless they’re granted special permission from Ottawa to the White House on catching high profile fugitives, even then: their powers are limited in order not to violate their sovereignty. Would their badge hold any weight while they are overseas (who have their own judiciary)?
That’s only speaking on official terms: but, what happens if an American on vacation (like off duty sheriff) witnesses or encounters criminal activity while they’re abroad? Are they allowed to detain that person or do they have to eventually turn them over to local authorities? For instance, they could flex a LAPD badge in Japan, but is that valid or considered null?
In Germany everybody, this includes foreigners, has the right to detain a criminal. But there are some things to consider. First you are endangering yourself by engaging with a criminal. Second if you are wrong and the person you are detaining hasn’t committed a crime but, lets say, just a petty offence, you yourself can get in trouble for unlawful detainment and assault.