Even traveling domestically, I almost always have my passport with me in my backpack, so I’d be fine there. I often have my passport card in a different spot in my backpack as backup, so there’s that, too.
When I travel internationally, I tend to only carry only what I really need on my person, and split it up into a few pockets - a bit of cash and a credit card in one pocket, some more cash and ID (usually passport card) in another. If I’m wearing a jacket I’ll stash a little more cash and another credit card in there. Passport stays locked in the hotel room safe unless I am certain I will need it for something. I figure that even in someplace like Rome, I’m not likely to get all my pockets picked.
I also print out (or e-mail to myself) the address of the nearest embassies and consulates so I’ve got those in the unlikely event I need legal help, or to replace a passport in a hurry. If I remember to, before i leave home I take digital copies of all critical documents (passport, birth certificate, air itinerary, etc.) and put them on a thumb drive that I put in the hotel safe as well. Not “official” ID, but I figure it can’t hurt in helping to establish who I am.