To be honest, I don't know. I would suspect that they are saved locally. I have read that the host is the only one with access to the save files, however.
Some things you should know before you play DS3 with random strangers, though...
When you play online co-op, you are playing one of the host's companions. You don't actually get to play your own character (it's like Fable 2).
This means that after you leave, someone else can join his game and play your character (as far as I understand). If the host decides to continue gaining levels, the other person playing your character will be able to adjust your stats after he levels up. I'm not sure if the person playing your character will be able to sell/destroy your gear, but the host will be able to. Not only that, but the host can decide to stop hosting indefinitely if he wanted to, which would mean you would lose your character (since, after all, you can only play as the host's sidekick). Characters created in a multiplayer game are not persistent across different matches hosted by different people.
I would highly recommend that you play this game with friends, or people that you know (like if you were to meet up with people on a forum and set up a game there).
I hope this helps. =)
Not exactly. As far as I understand if you rejoin you get your charachter back and he levels up to the host level if the host decided to play further.
Can't say for certain though as I've not tested online.