Only the pilot in the LHS can log P1 time as they are PIC. You are NEVER PIC unless they die or are incapacitated and then there would be a legal argument over whether you are really PIC anyway given that they signed the tech log and loadsheet but that is straying off the point. If you ripped the gear off on a diversion with them ko'd beside you I would certainly be getting my lawyer to argue that anyway
But anyway I digress....
The safety pilot in our airline won't be logging anything during your initial training. During line training I took the view that I was receiving dual training and logged it as such. Same for base training since I didn't have a type rating issued at that point.
Once I had been line checked then I started logging my time as P2 when I was PM. When flying a sector as PF on the line after my training was complete I would ensure that each skipper was happy for me to log the time as p1/s and had them initial my logbook. This was only for the purposes of getting the required hours for ATPL issue. There are rules about logging P1/s which are in LASORS. If the skipper takes control or overrides your decision making then you are no longer acting under their supervision as P1/s.
In reality should you go for another job then all your multi pilot time is either P2 or dual until you pass your command checks and can log P1. Up until that point your only splitting hairs and in reality keeping a tally for getting your ATPL. After that its plain sailing.
In practical terms I have one of those big black JAR FCL logbooks. I carried forward the totals of all my single and multi engine training into this book. That made it easy for me to distinguish at a later date for ATPL issue all of the various hours by a/c type, VFR/IFR,day/night etc etc. Trust me if you don't sort this out now (get the forms and have a look) you will spends days on excel trying to get it all to work and add up correctly. Its also a good idea to keep track of your airlines records and make sure that your logbook agrees with that. You may be required to provide to the TRE doing your ATPL skilltest a printout of your hours and this should agree to your logbook.
Above the P1 column I annotated in pen P1/s. In the remarks column I note whether I was PF or PM and things like Cat3 NO DH etc. When I am PM all my time goes into the "Co-pilot" column. When I am PF I put it under the "PIC" column which as I said I annotate with P1/s since I am not PIC.
If I was filling out an application form for another airline then all of my multi pilot time would be either P2 or Dual or F/O depending on how they wanted it split. The only real PIC time I have was when I was flying a single engine around hour building.