An update based on my personal experience:
We arrived in Rome last night (September 23) and took the train to Florence.
After jumping through hoops to get a Covid test within 72 hours of arrival in Rome and get the results back before my flight out of Seattle, NO ONE HAS ASKED FOR or looked at my negative test results. I thought I would have to show it to board my flight, although, since I had a connection in Frankfurt, most passengers on that flight weren’t going to Italy and didn’t have this requirement, so, I thought, that’s why it wasn’t checked. I thought it would be when I boarded the leg in Frankfurt for Rome. No. Nor when I arrived in Rome. I’m not complaining: I think it’s a good idea to require it, I wouldn’t consider taking the risk not to have it, and I would feel safer believing everyone coming from high caseload countries had had such screening. But I’m perplexed by the lack of enforcement. Maybe it’s assumed that no one would deliberately take the chance of sneaking by without the test.
My proof of vaccination was checked, both before boarding the first flight and on the train (although not before boarding the train). We’ll see how museum monitoring goes over the next few days.