SoCal App isn't informned properly. It does happen that people may not be aware of all the visa stuff for their own country. Or simply doesn't know. No offence, but..
aviation J1 was training and instructing experience visa for 2 years max. It was finished due to haggling of US DHS, DOS etc. Long story. There's sort of patch F1 visa now, but more limiting, done by those schools that used to have J1 rights until recently.
J1 is quite wide variety of purposes and different durations depending on use.
J1 is used for au-pairs (not flying), scholar/PhD/research Uni visa for up to 3 years (again not flying), Work and Travel summer up to 4 months work visa for Uni students from abroad (not flying again), then there's the most commonly used J1 besides aupair, and that's for internship, 12-18 months. And more..
So OP probably refers to internship/work placement visa. Since it's temporary for not too long, H1 or L1 class do not make sense, as these are more permanent and 'dual purpose'. J1 is strictly non-immigrant.
So it happens when you mention J1 on PPRuNe, everyone thinks of flight training J1 straight away..
Since you have legal status in the US and you are entitled to work (albeit for one employer who sponsors your J1 placement), I do not see any problem regarding training for PPL. Just like being on F1 or H1 visas, flight training is secondary and not tied to anything.
You still have to do the TSA paperwork (presumably to check that you are not a threat to national security blah blah), though. Don't forget that.
Your legal status is not tied to M1 that is most common for flight training, so you are quite free to choose the flight school and change it should you not like it in some way. As you can suss out, JAA PPL training in Florida is kinda overpriced and of not same standards across the schools.
I don't remember details about TSA bureaucracy for changing schools. You do provide your school details on TSA website, they notify them that it's true that you're about to start with them. Then once you do the registration, fill out all the boring details about your life, addresses etc, pay the fee, you get conditional approval that is finalised once you do the fingerprints. It's cheaper and mroe hassle-free to do them once in the US. Not much of a delay. In the UK doing it in Oxford is more expensive and you have to book ahead etc. Not worth it. I was going to but then the guy travelled somewhere. I am glad I only paid 30 or 40 bucks for fingerprinting in Portland, OR, instead of about 120-150 dollars equivalent. Don't remember now.
The approval from TSA is valid for a year, for initial (PPL) training, so from final approval it's one year to get it done. Surely not a problem if you are there. I haven't looked into what to do if changing schools, with TSA, but the worst case is you pay that 130 bucks fee again. With all the flight training expenses it's nothing. It shouldn't be such pain in the back like changing a sponsor/school while on M1.
The only problem was regarding status for some when they got over on visa waiver or B1/2 tourist visa and wanted to do rating/cert issue training, towards those. That's supposed to be on M1/J1/F1, not as tourist. As tourist, people can legally hourbuild on FAA revalidation or full FAA PPL etc. So long you comply with TSA, it's not an issue if you're on work visa in the US.
Check FAQ on
http://flightschoolcandidates.gov