Hi!
Virtually every airline and aviation organization will want you to log PIC when you are responsible and you sign ALL the paperwork.
So, all your solo time is solo...they won't even accept that as PIC!
Now, if you want to have two separate logbooks, or track it separately, that is up to you.
I decided to match mine up with how the airlines do it, so it was simpler for me. I would also STRONGLY suggest get an electronic logbook.
Another consideration, that would have made my appications easier, is that a lot of foreign places want/allow you to break out your P1 (us) time. P1 is PIC, P2 is SIC, and P3 is???FE, not sure as don't have any P3 time P1 (us) is PIC under supervision, so that is where the dual stuff would come into play. Also, if you are typed/rated for the aircraft you fly, you can count 1/2 of time as SIC as P1(us). All of time in formal upgrade (to captain) training is P1(us).
There is no P1(us) in the US.
cliff
NBO