The main implications are social.
True. I've been thinking of it as a kind of 'social scope' where you can decide on the size of community: one a large, sprawling metropolis [online], the other smaller, isolated pockets [local].
If your game expects people to get invested by learning deep mechanics or listening to a story, then local multiplayer is going to fight against you. People will be around their friends and less able or willing to focus.
Interesting way to put it.
But then: what of a local co-op game? Wouldn't that
engender immersion as players have to work together and are (as equally as they could be) invested? I think so and it's one of the aspects I've been struggling with. Even if the game is primarily centred around local multiplayer, I think the option of online is a 'nice to have' sort of feature.
What if you can't get friends and the game requires there be three players minimum? Should one care for that scenario as a developer (or just implement bots and call it a day)? And at what point does one have to 'give in' to online because it's a more 'scalable' architecture? I mean, supposing a game requires an absolute minimum of four players - and it only grows from there - it makes more sense to go online.
Rambling done.