the Pilot is responsible to fly the aircraft, so he is PIC and can log PIC and instruction received.
the flight instructor is responsible for the instruction. He is not reponsible for the aircraft so it can be a non FAA flight instructor and he will log flight instructor time (and PIC time, but PIC as a FI)....
I know, this is crazy!!!
this is more a problem about who is going to pay if there is an accident and in this case you should contact your insurance company..