Chester:
Regardless of which seat you are sitting in, and who signed for what, according to the FAA if you are the pilot solely manipulating the controls , whilst doing so you can log PIC time.
Technically no---you also need to have privileges to fly the aircraft legally.
14 CFR Part 61.51(e)
(e) Logging pilot-in-command flight time. (1) A sport, recreational, private, or commercial pilot may log pilot-in-command time only for that flight time during which that person—
(i) Is the sole manipulator of the controls of an aircraft for which the pilot is rated or has privileges;
The biggest issue would be if you do not hold an FAA license and do NOT have a jetranger type rating on your CAA---then you cannot log PIC time in the Jetranger in the US. You also cannot have two people simultaneously logging PIC unless one of them is a CFI.