I've never given up my passport - I refuse to do so. On seven journeys around Europe I've covered Ireland, the UK, Italy, and stayed in specific cities Paris, Berlin, Krakow, Prague, Budapest, Brasov and Bucharest. A budget hotel in Bologna, Italy held onto our passports while we had the room key but anytime we left the building, we returned the key and got our passports.
A hostel in Belfast insisted on an ID, so I gave them my driver's license instead. That's a $4 replacement online in Arizona and I have spares anyhow, so no big deal. At Garibaldi Guesthouse in Budapest, the proprietor held onto my girlfriend's license until we paid for our five nights in full the next day.