'Virtual' postdoc (in math)

In person and Skype are completely different. Writing math that you can both see becomes a challenge (in person, it's called a blackboard). I get up to fetch a book: communication is lost. Meeting takes longer than expected and it'd be nice to continue talking over lunch? Very difficult with Skype. More than two people in a meeting? Complete nightmare.

And a postdoc position is not just about talking with your supervisor every now and then. IMO, that's just a recipe for having a terrible time. You are expected to participate in your department's life. This means going to seminars, talk with people who you don't collaborate with, etc. Impossible if you are restricted to formal Skype meetings with your supervisor. Let's say you have a quick question about something you're reading. Are you going to call your supervisor about it? Or go to the office next door and ask, maybe even someone else, like another postdoc or a PhD student who works on the same kind of topic?

What you are describing is a research collaboration with someone else. You check up every now and then for status updates. It's not what most people envision when they think of a postdoc.

There is a BIG difference between in-person and via Skype. I have seen this before on several occasions in industry and it simply does not work as well. I doubt it'd be different in academia. It's much harder to integrate a remote worker into a group, especially if they're in a different timezone. I suspect there would have to be a very compelling reason for most supervisors to agree to this. Maybe they'll do it, but you better be pretty special.

I think that would be entirely up to whoever is funding and or supervising it. I doubt that many would wish to do so as there is little opportunity for direct supervision or keeping track of the activities of the post-doc. It would seem to depend on an inordinate amount of trust.

If you were the funder, you should probably establish some very regular means of communication to see that your resources aren't being wasted.

Such a thing might be more reasonable if the post-doc and the PI had a regular relationship in the past so that trust was already established.

I don't think there is anything wrong with such a thing, I just don't guess anyone would be interested in supervising it. Interesting, but unrealistic.