For starters, people would want to be able to take characters played in offline mode back onto the server. That's a huge, HUGE opportunity for people to cheat. How many times have you played Call of Duty (a peer-to-peer game at its heart, even with Steam matchmaking) and run into somebody who obviously hacked their level to unlock all the perks?
"Well, just don't allow characters played offline to rejoin the server" you say? Hah, good luck with that, especially if your MMO is pay-for. No matter what anyone claims now, I promise you that if it were implemented that way (offline characters couldn't sync progress), it would get rightfully reamed by critics and you'd hear no end of complaining about it.
I don't claim to have answers that are guaranteed to work, but I do have answers. The model to reference are console games that have a single player mode and a multi-player mode. These are considered completely different experiences and the expectation is set that unlocking things in single player does not necessarily get reflected in multiplayer. You'd treat the single player mode as a sort of "practice mode" for the multiplayer mode.
But more specifically, in the example I gave above the design wouldn't be separated into "single" and "multiplayer." Those would be two extreme ends of the spectrum. Sitting in between "in single player you have ultimately full control" and "when you log into the big server they control everything" there would be the peer-to-peer intermediate level of play. Here, I envisioned a chain of trust model that is similar to, say, PGP keys. If I trust you and you trust me, my game would allow you to import your characters into my instances and vice versa. We'd allow that, because we get to decide what happens on our own instances. Webs of trust could be formed whereby who gets to import what is arbitrated by trust levels and ACLs. Maybe I trust all of your character progress, but none of your powerful items. So when you "transfer" to my server you keep some stuff and (temporarily) lose other stuff.
Its how PnP games have been arbitrated for years. There was an honor system involved that was enforced by the simple fact that in small gaming circles cheaters were risking being ostracized. Peer pressure was the way to ensure cheaters didn't abuse the system.
I was actually thinking about this very model, on a smaller scale, with regard to the Arena. One of the things I suggested to the devs was that rather than make the rules of PvP carved in stone, at least in the Arena allow players to choose their own rules by consensus. That way the participants on an event by event basis could decide if they wanted to play with movement suppression, PvP diminished returns, heal suppression, etc. Give players the ability to set "profiles" for PvP and have the game have a system where those are arbitrated among players joining an event.
Multiply that by, oh, a hundred, and you have a distributed peer to peer "house rules" system of what to allow individual players to use in other people's games. In the big central servers the only things you'd probably be allowed to bring are the literal shirt on your back: you can transfer character appearance, but all items and progress must be earned on the server. So when I play there, I'm "exemped" to what the server thinks I've earned. On my home server, I'm back to level 50. When I play on your server, I get to keep my level 50 because you've set that level of trust, but I cannot bring my full complement of incarnate powers unless I earn them there first, because you trust me, but not that much.
How to make the economics work is a completely separate issue. Its related to the question of "why play on the big servers at all?" And the answer is, the operators of those servers have to make it worth people's while through introducing worthwhile content and worthwhile overall player environments. Not easy, not impossible, but probably a similar nut to crack as how to convince people to keep buying fruit bonuses in Candy Crush.