We are renting an apartment in both Barcelona and Nice for one week each. If we feel the apartments are secure, is there any reason we need to carry our passports as we sight see, rather than leave them at the apartments?
Your passport is probably more at risk walking around unless you plan on wearing it in your money belt and that can get more than a bit old. I mean, don't leave it out just anywhere, say put it in a box of cereal. You should always carry I.D. however. Europeans carry identity cards so bring your driver's license and a photo copy of your passport (optional).
A photocopy of a passport has no value as an id other than prove to the US that you once had a passport should your's be lost or stolen. You donot use a photocopy of a passport as a substitute id. I always carry my passport as it is the only thing that proves I am legally in the country.
Immigration is a big issue in Europe (as it is here). Your passport proves your ID AND that you are in the country legally. You more than likely won't be asked for your passport, but you can be fined for not having it with you if you are asked.