I can only give an opinion based on my own experiences but, I'll try.
I spent most of my life in the US, a brief period of time in the Persian Gulf, a couple years in Southeast Asia and a few months in the UK. I have never experienced anything unsatisfactory with the Italian healthcare system other than 'creature comforts' and the possibility of long waiting lists for specialist exams. The pediatric staff and others I have encountered all seem extremely competent and give no indication of being less professional or capable of caring for patients than American medical professionals. Hospitals and doctors offices are always well-equipped with necessities and clean, but the food is skimpy and it is not luxurious in terms of large flatscreen tvs etc (at least south of Rome, anyway). When I go to an airport, I am more concerned with the competency of the pilot and integrity of the aircraft than how attractive the duty-free shops look and how comfortable the chairs are.
It seems Italian medical professionals are less pushy with pills than their American counter-parts; perhaps they don't get as many free lunches or other sorts of incentives.
If you go to the hospital, you don't get a bill in the mail. However, some routine care is not free unless you are below a certain income level or have a chronic disease and can get a 'ticket esenzione''. So, for something like an outpatient blood test you may end up paying in the area of 70 euros and you may pay roughly 25 to see a specialist. Most drugs are cheap from what I know but I cannot offer in-depth advice on that.
As far as private care, I don't know because I have never had it, but have heard mixed reviews and gather that the main difference is convenience and comfort rather than actual competency of the professionals.
I do believe that American medical professionals earn way bigger salaries than their Italian counter-parts, but I get the sense that the Italian ones are motivated more by a sense of purpose--that it is more about pride or a passion in life than a paycheck.Â
I do not want to cause anxiety but feel that I'd be wrong not to mention that recently, in a town called Cisterna di Latina, there was a Filipino child who was rushed to Bambino Gesu' hospital in Rome for some vomiting and headaches or something or other that turned out to be a rare incurable neurological disease, he was discharged way prematurely IMO and then went into convulsions at home and died in the helicopter on the way back to the hospital. No medical system in the world is perfect or can prevent every tragedy, but I'm just saying don't let any hospital send you and your child home without undergoing at least 24 hours of observation and absolute reassurance that what happened at Bambino Gesu' with the Filipino child won't happen again.