The American Mafia and the Italian Mafia are completely independent organizations. Obviously, the American Mafia is a descendant of the Italian one. But there is not real direct connection. La Cosa Nostra is a homegrown phenomenon and doesn't answer back to Italy. They do however collaborate together for international rackets and where there are other links they do business.
On another note, there are Italian abroad working for the LCN. The most famous example of course is, like Nucky said, The Cherry Hill Gambinos (Sicilian faction). I would think they are given a degree of autonomy withing the Gambinos but naturally have to answer to the Gambinos while they're in America.