no matter which fliying club you fly in NY or San fransisco...Your license will be FAA one. The only diffrence between diffrent schools is the quality of instruction, and the cost (also schedule, aircrafts...)
now , if you do it outside the US, it's another thing....
Note that you can have a JAA private pilot in the US also (EFT or EASA for exemple)....
Hope that's answer your quiery