From a role-playing perspective, there seems to be nothing to prevent spellcasting players from teaching other characters the Magic Initiate feat. The subject of players receiving training in lieu of financial rewards is addressed in the DMG on page 231, under "Marks of Prestige":
A character might be offered special training in lieu of a financial reward. This kind of training isn't widely available and thus is highly desirable. It presumes the existence of a skilled trainer — perhaps a retired adventurer or champion who is willing to serve as a mentor. […]
A character who agrees to training as a reward must spend downtime with the trainer […] Possible training benefits include the following:
- The character gains a feat.
I can see this being objected to from a meta-gaming perspective, but from a role-playing perspective, it seems logical that experienced characters who've adventured together and rely upon each other to stay alive might want to teach each other feats in order to increase their collective chances of survival.
Can player characters teach each other feats?
Best Answer
First things first
This Training sub-section is under the Other Rewards section. Read it carefully:
I.e., this is obviously assumed to be a reward given by the DM, not a player training each other. Using it that way is not intended.
Role-playing wise, the easiest explanation is: PCs aren't that skilled. This might be a hole if you are talking about 11+ level characters, but check the description of the trainers:
None of these fits a low level character. The bard in a low level party is barely known in the local inn. The Barbarian is now starting to get used to brawls. The Wizard has 10 spells on his one book, not a tower with a library for himself. They are not the high skilled trainer that would be able to train others.
As for the balance
While I see KRyan's point, I don't think training is that problematic, balance-wise, if well handled. Feats are on par with +2 ASI, which is what you get from items like Tomes and Manuals - you may reward a feat when you would reward one of these tomes, instead, for example.
As mentioned by András in the comments, yes, some feats are stronger than simple ASI, but IMO, that's a problem with the feats balance to start with. The players will be getting these feats, you reward them or not, unless you are not allowing feats in your game. Rewarding the feat instead of the tome simply changes the order when they get the feat and the ASIs that they would get as well.
Final note
Also, it's worth noting that this section of the DMG is more of a "inspiration/ideas for you" than actual rules. Note, for example, the lack of how many days it would take to train, how much gold it would cost (if any at all), or anything telling us what would be a good actual reward. What I mean here is: don't take this part too seriously without thinking if your campaign needs it or if it is able to actually support it.