In essence you must be qualified to give instruction in the State where your wish to give it, assuming the aircraft is also registered by that State.
You cannot legally send a student solo unless you are qualified as an instructor by that State.
You would obviously need the JAA qualification as well, and the right to work in the US; the more difficult bit!
Some JAA commercial instructors work in the US on the basis of a UK/JAA commercial licence, with a FAA (Restricted) validation and do not hold a FAA CFI rating, as there are no solo sign offs on the JAA modular commercial. It does however depend upon the interpretation of the rules by the local FAA Flight Ops Office.
As you would need a sponsor to get a Visa, ask them!
Keygrip:
"I know some JAA guys have been teaching nothing"
I think I have seen some of their students!!!