I'm not sure those who are being totally dogmatic on this issue are correct. I was told something different by a flying club in the US, and it made sense to me. If you are in the US for another purpose (eg business or visiting family), then without a visa (but with TSA approval) you can do some training while you are there as long as it doesn't exceed a certain number of hours per week. I don't remember what the number is (15 rings a bell but don't quote me). The key is that you are not there solely for the purpose of training, and that the training is incidental to the purpose of your visit.
Anyhow in 2012/13, over a few trips to visit family I accumulated enough hours to get an FAA PPL, and had no trouble with the IACRA system or getting the FAA examiner lined up. Your experience may differ of course, etc etc!
I think it is in reality a bit of a grey area. I would say that obvious abuses are likely to get you in trouble, and there may well be a distinction between a flying school offering a PPL course vs a flying club where you make a private arrangement with a flying instructor to get yourself up to the required standard.