I think things have changed drastically over the past few years in this area. At one stage it was very tight, but now, a CFI at a JAA school can recognise the hours for any ICAO training as counting towards a JAA PPL course.
This allows someone to part-train abroad in another non-jaa country (eg: S.A., Oz) and providing it was for the ICAO PPL, these training hours can be recognised at a UK school and the JAA course completed there. (It used to be a maximum of 10 hours could be counted across to the 45). Why is this relevant? It allows the US JAA schools to employ FAA instructors with FAA only qualifications and then the school's CFI can 'recognise' the hours within his/her own school done by non-JAA instructors.
Some might say 'so what?' but I do know someone recently back from a Florida JAA school who was concerned that all his instructors knew about JAA was how to spell it, so they couldn't answer his questions on it. Some might say 'so what?', but he found this worrying as he wasn't sure if there was anything else they didn't know. And - he had a big turn over of instructors as the school were employing 'newbies' and then 'letting them go' for fairly minor reasons 'pour encourager les autres' knowing that there was a queue of newby FAA instructors wanting jobs and hours, 'pre-airline'.