Hi
As an older Military ATC, I think that makes more sense to make a training and "working" for each Airport. In reality, after we get the ATC Course, we are transfered to a unit (Airport). Then at this Airport we are trained, after the successfull training, we can take control of the position without beeing monitored. London Control for instance has this type of concept. First we are trainee, after some hours without errors (or big errors like lost of separation) it gives us the possibility to "work" on the sector. If we make big mistakes we will be "sent" again to trainee mode in this airport.
So even if we are not trainee in KLAX for instance, if we want to control at KSFO we must pass the trainee sessions for that Airport too. And so on...
Regards
Paulo Moreira