In Italy they register you with the police and need passport data. I never leave my passport at the desk for more than an hour and preferably not at all. If they can't get it done immediately, I leave a copy which I carry for this purpose. Years ago in Siena I came down for dinner to see our passports casually left on the counter in an otherwise empty lobby. Anyone could have picked them up. No one else cares as much as you do or will have the same hassles you will if the passport goes missing.
We always carry a copy on our person and have the original locked in the hotel safe. We had some business at the US Embassy in Paris a few years ago and the waiting room was full of tourists in a swivet because their passports were lost or stolen and they needed replacements for onward travel. The woman next to us had missed her flight to the next stop because of this. And everyone was wasting a day dealing.
In 55 years of international travel I have never been asked for my passport except at moments when it is required i.e. border crossing (not in Schengen), banking and purchasing phones or sim cards. Even VAT just requires a copy when getting paperwork in the store. I figure if I actually having police demand my 'papers' the least of my problems will be obtaining it from the hotel safe.