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, anyone has the right to provisionally detain a perpetrator caught in flagranti until police arrives. This would also apply to the US sherrif on vacation.
And this would also create an incredibly dangerous situation for everybody.
The Us cop would act according to his own (non-German) training and daily routine, which includes lots of violence that he uses (because he would have the right to do this at home) on the suspected person and maybe on other persons standing in his way. The US cop would also expect and fear violent reactions. But people would react very differently. He would surely misread most people’s reactions and then probably escalate his violence far too much, and far beyond all that is allowed for such an “everybody detainment”. In the end he could be the only one charged with crime, and pay for other people’s hospital bills.