I think the safest answer is that it can depend on the person selling tickets, so if you want to be 100% sure, it would be best to have their passports--though a photocopy would probably be fine for this purpose at museums, etc. The odds are that if the kids look young, you'll never be asked for proof, but carrying a copy of the photo page of a passport isn't burdensome, so that's the middle ground I would choose.
I'm a senior, and I have not been asked to prove my age except when buying or using senior rail passes or tickets. For that, a passport (and not just a copy) is fairly often required.
I'm basing these comments on my experience in Europe in general, not Italy specifically.