The same works with other types of units too. You just change their training mission. You can train Army pilots in Army units, USMC in USMC..and so on. Logical.
You can even train carrier pilots with short range USN floatplanes. And patrol plane pilots. Only Torpedo bomber pilots are hard to come by since most planes cannot carry them. Thus no torpedo training...but there are remedies for that. Either you pull out front line torpedo plane unit for training and remeber to check "Use Torpedoes"..or even more preferably..you get those CVEs with VT unit and use that. I don't remember if later war CVEs with VR-units (VRF, VRB, VRT) as replacement units come with torpedo bombers, but if they do, use them to train torpedo pilots.
Another thing is..pilots are not tracked for Carrier Capable/Trained, only units are. So, even when your reserve pilot has been trained with OSU-3 Kingfisher float plane and never been abroad CV before, he can if he goes to unit that is Carrier Capable/Trained. Bit of abstraction there.
"To meaningless French Idealism, Liberty, Fraternity and Equality...we answer with German Realism, Infantry, Cavalry and Artillery" -Prince von Bülov, 1870-