The US and Italy (like many countries) have a Tax Treaty which prevents double taxation. If you are a resident of the US, you do not pay Italian income taxes. If you reside abroad in another country the rules are different but I've never lived elsewhere so am not sure how that would affect things.
It has since changed, but when I lived in Canada, I was expected to pay any money my taxes to Canada would net out against filing in the US. Or something like that.
Thanks for the info. I always wondered how that worked.