I can't speak specifically to any local fertility clinics, but I can tell you a bit about what you might expect from Italian medicine vice American medicine.
The NAVHOSP can't give you recommendations, which I know can be frustrating. However, let me say that 'private' Italian clinics are known for being very good (I've used one in Rome and one in Naples). They are good because they specialize in their area - that is all they do, day in, day out. I've also found them to be decidedly more thorough.
They will likely expect payment up front. Best to inquire in advance if they will take your insurance. If they will accept your insurance, you will need to bring in a Guarantee of Payment (in Italian) from your insurance provider. If being paid out of pocket, you will need to pay up front, generally prior to or immediately following your appt/procedure.
Be prepared that they may not accept your US test results (they aren't translated into Italian and they need to have their own understanding of your health). Being as thorough as they are, this is a positive thing. If their results differ from those in the US, don't immediately discount the Italians - chances are, they could be right.
Bottom line, you need to do what you feel is right for you. Go visit the clinic if you get the opportunity. If it is clean, efficient and busy, then there is likely a reason for it. Have you ever heard it said that you should only eat at a restaurant if it is frequented by Italians, because you know it will be good. The same can be said for medical care. Good luck!!!