However, I have a photocopy of the passport "just in case", and keep it in a separate place and/or provide a trusted friend at home with the copy, that could be e-mailed or faxed to me if my passport is lost, so as to continue travel or expedite an emergency replacement.
I carry a photocopy but also upload a pdf to secure online document storage. That means I can get the passport copy from any computer in the world with internet access.
As to the OP's question, I don't carry truly valuable stuff when I travel. I might leave my keys and US-only cell phone in the safe, but would leave anything really valuable in the front desk safe.
I would also accertain, in advance, if the place has a front desk safe before booking the hotel if I had a need to secure very valuable items.