While police might be annoyed by this, you may be unlikely to encounter a situation in which it's necessary at all, and it probably won't endanger you unless you're in a place where police habitually try to extort foreigners, in which case it could provide a new pretext for extortion.
The best advice probably varies by country, depending on local crime patterns and also on the attitudes of local authorities to visitors (or to visitors like you).
The biggest issue I had when I got my passport stolen in Costa Rica was not getting my British temporary passport reissued, that was relatively straightforward. It was waiting in line to get an ESTA to transit the US on the return flight. Felt sorry for all the local guys in line to get their document processed. It was annoying because I didn't want to leave the airport in Florida or wherever I was transiting.
I remember in Nepal kayaking and basically taking all of our valuables on my body while on the river as we weren't sure we'd ever see our bus/driver again. This suspicion was actually proved valid when our bus was subjected to a minor terrorist attack!