Only general and training do.
This applies to Fleet training; for Crew Training, General and Naval provide bonuses. Looking at some of my units that went straight to a Naval command (i.e. fighters and FACs; everything bigger gets a turn in Training command first), it does appear that their local commander does need to have some level of crew training to make any progress - I have units that were built at the same time with significant crew training (22%+) and others with 0% that were built at the same time; both have usually had a commander of some sort, but for fighters I prioritize Fighter Combat over Crew Training.