No plans/ Passports

17 and 21? Wow, danger age - "If Mom says X, I will do Not-X".

I'd keep the passports in your secured belt.

We’re traveling with our 2 kids ages 17 and 21. I Don’t know if I want to leave our passports in our apartments, when we’re touring. Should I have the “kids” keep their passports on themselves (split them up) or should I keep all 4 of our passports together in my secured belt?

Definitely in your secured belt.

A 21 year old is an adult. If they aren't capable of personal responsibility at that age, then when? The 17 yo- you know his level of maturity better than anyone here. Give each of them a money belt or neck wallet. Discuss with them the importance of keeping their documents secure, and the consequences of carelessness.

What CJean said

Just keep copies in your hotel room.

I know technically you’re supposed to carry your passport with you, but…

Over many months of leisure travel in Europe I’ve never once needed my passport on an ordinary sightseeing day. Just for hotel check-in, rental cars, and air travel. Even when I needed to leave an ID as a deposit for an audio guide, I was able to use my drivers license.

in italy at least you are supposed to carry id at all times. while it happens infrequently, i have been, and seen others being stopped by the police at train stations asking to see id. just a few weeks ago the police actually got on the train and randomly asked a few people to show their documents. not me, but i always have mine on me. don't know what happens if you can't show id or if a photocopy/photo suffices.

I'd book the Doge's palace on the official site when your plans are set. Why waiting in the tickets line with two bored "kids"?

if I want to leave our passports in our apartments

You can't. Or, better, be aware of the risks you'll be running ignoring local laws. You are not only "supposed" to follow the rules, even when chances of passports being checked are close to zero.

The 21 years old is an adult and cops will treat him/her like an adult. If his/her passport is in your money belt, they could make the kid wait in a Police station till Mom picks up the phone and comes to the rescue with a valid ID. Imagine this happening at 3 AM...

I rarely carry my passport in almost any country I visit. I do have my driver license in my wallet. If I were visiting Iran, Myanmar, or China, I may carry my passport. But, Italy. Nah.
I agree with what CJ said about a 21 year old being an adult, but it only takes one person in a family to misplace or lose their passport and the whole group is involved and disrupted. It will not be only the problem of the passport holder.
I have never been asked for my passport outside of an airport in any country.

I would carry a photocopy of the passports while out and about, but keep the originals in the apartment. I've heard many times of pickpocket stories, but never of theft from apartment/hotel room.

https://community.ricksteves.com/travel-forum/italy/carry-your-passport-and-get-idp

What Paul-of-the-North said.

While the 21 year old may technically be an adult, having the "worldly" experience of an adult is a completely different thing. And the 17 year old is not even close. If either misplaces their passport, your vacation mode instantly becomes recovery mode.

Would completely agree with booking any tickets you can ahead of time.

My 20-something daughter had to produce her passport on a train between Rome and Orvieto last year. At a station stop police boarded the train and went through the car methodically asking each passenger for their "document". Not sure if this was a random check or something targeted, but it does happen.

First of all, the 21 year old...of course this adult should carry their own passport.

The 17 year old is almost an adult and needs to be prepared to be an adult; have he/she/they use money belt with passport or secure bag. If the 17 year old has never had a responsibility like this, then I would keep an eye on the money belt/purse/whatever from a distance when off the 17 year old's body...without them knowing.
-These kids are essentially grown (well one is) and need to suffer the results if they lose their PP, etc. while you are there.
-I'm one who carries my passport. It could happen that I can't get back to my hotel and I want to be able to leave the area should something unusual happen.
-I would also make sure they had an emergency credit card and a little cash each in case you are separated.
As a result of teaching adult skills, by age 18 my daughter was traveling in Europe with fellow 18 year old, independently. I taught her these skills earlier in her teenage years so she learned to help keep track of her things and be able to take care of herself. It's really hard to let go, but we must do it.

Sorry if I sound harsh...still drinking my morning coffee.

I doesn't matter what we think about a 21 and a 17 years old. I doubt the kids will spend the all vacation within Mom's reach: ER staff and cops will treat them as adults. Even if the chances of ID checks and medical emergencies are close to zero, "close to" is not zero.

Whereas zero pickpockets are interested in stealing a chipped passport that's worth nothing to them. It's just an evidence of a criminal offence.

Anyway, now krissycahill knows the (minuscule) risks they are running and the choice is up to them. I'm sure it will be a great family vacation, passports or not.

I assume the kids will be carrying their own cell phones. If they are responsible enough to be entrusted with hundreds of dollars worth of electronics it seems to me they should be responsible enough to look after their passports.

If there's something you really want to do, you should book tickets ahead of time.

What Wray said.

My now-18-year-old daughter carried her passport in Spain last year and will be doing so in Greece next week. My now-28-year-old daughter did the same as a teenage traveler.

They are both independent, responsible, and comfortable striking out on their own in unfamiliar places and have the foresight to create backup plans in case we are unexpectedly separated or things go askew -- because they were expected to be independent, responsible, and comfortable on their own.

What a gift.

"Whereas zero pickpockets are interested in stealing a chipped passport that's worth nothing to them."

Whereas, after they steal the purse/wallet, they chuck in the trash all the stuff that's not worth it to them. Happened to me. They kept only the cash, and an honest person found the rest on the ground and returned them to me via the US Embassy!. Luckily my passport was in my apt and they "only" got my cards and cellphone.

Make copies, leave a set at home, and a set at your accommodation.
Carry the originals with you.
Up to you who carries their own, but it must be carried in a moneybelt or similar, under their clothing.
I have been to Italy many , many times, and have been asked for my passport a couple of times, so it does happen.
Plus, it is your ID when you are out.
If your accommodation has a burglary, or burns down…drastic example….how are you going to get out of the country and home without your passports?
Make copies of your travel documents and store in secure computer files as well.
Ask your kids what they would do, and get them involved in your travel safety plans!

after they steal the purse/wallet

Well, do not keep your passport in the purse/wallet. What about keeping it around your neck, like a press pass or something similar?