Neither. I think he’s looking for someone with a spare room in NYC where they can both come and go as they please.

So on any given day there are four possibilities both in NYC, both in SF, OP in NYC && roommate in SF, and roommate in SF && OP in NYC

It’s a pretty interesting concept, not one I’ve considered before, but one I imagine there would be interest in.

Don't see why it can't just be a situation where you have a different roommate in each state.

Cost vs privacy / alone time.

Roommate in each state allows more spreading of the cost, but it's more likely that you and one of the roommates are in the home at the same time. Presumably each of those roommates would be treating each home as their primary residence.

With the originally proposed setup, there's a fair chance that you are in SF while the sole "roommate" is in NYC and vice versa.

Yup, that’s it exactly. There’s a good chance that you would be alone in one city or the other. And in theory you could both treat the second room as a guest room. With consent, etc.

Yeah, I’ve tried posting on housing swap web sites. But it seems they’re more geared towards full swaps which are shorter.