If John offers to pay for Jane's PPL flight training, and Jane accepts it, is she violating rules against compensation? If so, is it feasible to get some kind of waiver? Does it matter whether or not John also acts as her instructor?
In my opinion, this is possibly legal, but depends on the totality of the circumstances. From the Hancock interpretation (emphasis mine):
An item of value is not compensation under the regulations, however, unless the pilot's receipt of it is contingent upon the pilot acting as pilot in command of an aircraft. If you loan your aircraft to a private pilot who pays the expenses associated with the operation of the flight (e.g. fuel) and you are placing no obligation on the pilot (e.g. ferrying your aircraft to a specific location), then it is unlikely that the private pilot would be considered to be acting as pilot in command of an aircraft for compensation or hire. We note that whether a private pilot is receiving something of value in exchange for acting as pilot in command is determined on a case-by-case basis and depends greatly on the purpose and objective of the flight.
If John makes a gift to Jane of $10000 to be used for flight training, with no other restrictions, John is not requiring that Jane act as PIC (as she can use it for dual training or solo, at her discretion), and is not gaining anything in return. As long as it is a bona fide gift and not some attempt to end run around the regulations, it is not against regulations, in my opinion.
The last sentence in the quote is important, though. If, for instance, John stipulates that the flight training must take place in his blimp, which he puts advertisements on, this is definitely not legal. Nobody can say whether this situation is legal or not without all the details.
John acting as the flight instructor makes this a great deal shadier. In particular, as CFI John would have a great deal of control over the flights flown, and John is receiving something of value in return (time building, if nothing else). Dual instruction may still be fine, but John should be very wary of paying for Jane's solo flights, since as her CFI he will be placing a great deal of obligations on Jane as the pilot.
It's completely legal because it's a one way transaction - provided the payment is a genuine gift with absolutely no strings attached.
Private pilots are forbidden from receiving "compensation" - i.e. payments in return for services. Jane is not providing any kind of service in return for the payment, so there is no violation of the rules.
Now there might be a suspicion that this is some kind of under-the-table deal where Jane is in fact providing services in return for the gift, but if that's not the case then there is no problem with legality.
My evidence for this is that it happens all the time. Plenty of trainee pilots accept gifts from family members for their tuition, and if it was illegal then people would be telling them.
I had to think quite a bit about the issue with John being the instructor, but I'm fairly convinced it isn't a problem, because John still isn't getting any benefit from the deal. (He's getting flight time, but he's paying for it so there is no benefit to him, because he could get that flight time just as cheap by just renting the plane a flying solo.) Because of this Jane still isn't providing him with any service. The situation would be even clearer if instead of giving Jane money to hire him as a flight instructor John gave Jane his services for free and payed all the expenses of the instructional flights