I dont know about sweden, but here in Norway(i think its the same in all JAA countrys) you can only fly on FAA license if its commercial. But there are some limitations. Only VFR day(so no ifr) and only fly in the country where the plane is registered.. If you only have a private, you have to convert it to a JAA private pilot licence thrue a flightschool. Oh.. You can only fly for 1 Year on your license from the first logged time. Im actually doing it now before my ATPL connversion