Not to start a flame war, since this is a very "popular" topic, but just putting this out there. I watch a British vlogger who lives in Positano (cleverly named Nicki Positano) and she was driving home for the holidays. As she neared the French border the Italian authorities stopped her and asked for her documents. She and her husband in the car with her indicated that it's not unusual these days for the Italians to stop cars and set up checkpoints, it's wasn't clear why but my guesses would be a) drugs and b) immigration.
She also didn't indicate what documents she handed over but keep in mind that if you don't have an IDP, you could be in for a long roadside stay. Cops out on their own have quite a bit of discretion and can interpret laws in interesting ways.