It varied by airline, but I believe that ALPA wanted all cockpit personnel to be qualified pilots but that F/E did not need to be qualified mechanics. I think that ALPA were happy for F/Es to retrain as pilots and to grandfather existing F/Es who couldn't but that all new entrants had to be pilots. The F/Es were not always happy with this and there were many inter-union disputes. Western airlines sacked all their F/Es sometime in the 60s during one. I believe that one airline (Eastern?) was proposing to employ jets with three pilots plus a F/E to try and keep everyone happy. In Skygods Robert Gandt suggested that all new Pan Am cockpit staff (in the 60s) were pilots but that the F/E contract was negotiated by the FEIA. If you joined as a pilot you could be assigned as a F/E and be paid much more than a navigator (S/O) which has always been pilots under Andre Preister as the F/E scale was much flatter and you were better off with low seniority. Someone may know much more than me. It was suggested that ALPA liked F/Es to be referred to as Second Officers to make it clear that they were certainly not equal to F/Os although another poster has said that qualified F/Es could still be referred to as such. Of course outside North America most airlines thought that having a qualified F/E on board made sense as they could sign an a/c as ready to fly in case of a diversion to a remote airport. It also meant quicker promotion for pilots to captain. Its interesting to look at what happened to F/Es outside N America when glass cockpits arrived and their role did disappear. Fortunately it was generally handled without too much rancour but posters might wish to say more. All of this is based on what I have read rather than actual experience so I'm very happy to be corrected.
I'm told that the reason that BEA did not employ F/Es was that they didn't want a second union involved - I can believe this looking at their industrial relations record.