Separate bills at restaurants

Yes.

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Asking the wait staff to split the check for you can be done. However, put yourself in their shoes. Say there are three of you, one pays by credit card and the other two pay by cash with larger bills and require change. It would be a big hassle for any wait staff regardless whether they are in America or Europe.

Our group of six used an expense splitting phone app called Splid. We all took turns paying for meals, attractions, hotels, etc. Splid keeps track of all the expenses and can automatically calculate how much the parties need to settle up with each other at any time. We would look at the balance from time to time and whoever was behind would make a point of paying the next bill. We racked up over €10,000 in mutual expenses during the trip. In the end, we only had to settle up of about €30 to 60.

File this under: "Things that are exactly the same in Europe as they are in your home town, wherever that is."

Can you do it? Yes.
Do people do it all the time? Yes, probably (depends on the restaurant and the typical clients).
Will the server be happy doing it? Probably not thrilled, but probably used to it.

Since you pay at the counter or bar in Italy, each person just tells the staff what they ate, and it's rung up. The trick is for nobody to forget anything.

As Bets said, Italians (including myself) pay at the cash register on their way out of the restaurant, therefore you don’t ask the server to do the splitting or anything for you. The person at the register (in most cases the owner of the restaurant or the owner’s family member) will handle the transaction and s/he will not have a problem doing any splitting of the bill. I’ve done it (rarely) and never had an issue. When you pay at the register you actually won’t need to remember what you ate in most cases, since the order (“La comanda” in Italian) will be available at the cash register desk either in paper or digitally.

Servers in Italy actually don’t really appreciate having to bring you the bill at the table. It’s extra work for them taken away from their busy activity and focus, which is bringing food to the tables. Italian restaurants are often understaffed because they have to be paid on salary at a level determined nationally by the labor unions’ bargaining contract. They are not paid minimum wage like in the US where most of their income is from tips. Therefore if you can spare some of the busy work for them, like bringing your check at the table they will certainly appreciate it. Walk to the register and pay directly to the owner, like the locals do.