Under FAA, even the flying time you are "under the hood" with an instructor you are PIC.
If something was to go wrong, ie an engine failure, you would still be responsible for any action you might need to take, at the end of the day you have all ready have a PPL.
It is very different compared to how things are done under JAA/CAA, where they wouldnt accept that.
It is all down to wether it is an N reg aircraft or European registered.