Logging Time
I believe that you can only log that time (as PIC) if it was actual instruction of the private pilot (i.e. "dual" in what would othwerwise be a single pilot machine). And for it to be instruction, you would have to have a JAR FI qualification to be training on a G registred aircraft. Your FAA CFI may allow you to train someone on an N registred aircraft; but be careful to check the regs on whether remuneration is allowed and if a special dispensation required.
TT