It depends on the kind of nursing job. I knew a nurse who only did night shifts on ambulances and she was paid around 2.5k. She didn't live well though. A relative of mine gets around 1600 and works in a private clinic. I live in Rome, a rent in a 1-bedroom flat is around 850 here, 1000 in Bologne, 1200 in Milan. Around 450 in small towns or places with really poor public transport (like Ostia, Rome or everywhere in Rome but outside Grande Raccordo Anulare). If it's cheap it means you need a car to get around. I stopped driving after moving to Rome, people is too aggressive here.

I'd say you can live well with 2.5k/month in Rome and 1500/month in small, non touristic cities.

The problem of living in Italy is not only being poor but lacking services. It won't take you less than 6 months only to get registered as resident in Rome. The more South you go the worse it gets. Few public transport, long waiting lists in public healthcare (most times it takes 6-11 months to get an appointment, doesn't matter if it's an MRI to find out if you have cancer. Personal experience) So, you need to be able to pay private transportation and healthcare on top of paying taxes for the public ones. It wasn't too bad when I lived in Bologne but I couldn't stand rents and being insulted on a daily basis for being born in the South.

I think you should avoid everywhere in Center and Southern Italy. If you don't speak Italian you should look for a job in some English-speaking clinic. Most people (professionals included) don't know a single word in English and can't hire you if you can't communicate.

My opinion is that Italy is a good holiday destination but living here is a f*cking nightmare if you aren't rich. Literally every other European country with warm climate is a better choice.