For CPL and Multi I would much rather believe a school that told me I did need a VISA and TSA, it's not that difficult, save yourself a lot of heartache and just do it.
A good school that trains a lot of international students will be well set up for providing the paperwork for VISA and TSA.
I know of some places where you can just turn up with a visa waiver and hand over the cash for training, chances are you would get away with it, but is it worth the risk...?