As far as I understand it. There are many issues here.
Firstly the rating will either be FAA (USA) or JAA.
The training organisation will be authorised for either one or both or other authorities, regardless of where it based. Hence you can do a JAA TR in the USA (provided a JAA TRTO).
The main difference between a FAA TR and JAA type rating is that under the FAA system they do not do the base check (take offs and landings in a/c) as part of the course. Hence FAA prices may be cheaper.
You can convert a FAA TR onto your JAA licence once you have 500 hours on that type. I don't know the proceedure for the other way around.
Hope it helps.