Thank you "I" in the sky, good post
The FAA has everything to do with JAA training in the US.
- You fly in a US registered airplane
- You are trained by a US certificated Instructor
- You fly in US airspace
- You solo on a FAA student pilot certificate and medical
- You require FAA instructor endorsements for solo flights including cross countries.
You cannot be trained for the
PPL by a JAA instructor only, the instructor needs to hold both JAA and FAA instructor ratings or FAA only if they work under the supervision of a JAA Chief Flight Instructor and have been standardized.
You cannot be Pilot In Command yet since you don't know how to fly an airplane. The instructor is Pilot In Command and since it is a US registered airplane your Instructor needs to be US (FAA) certificated.