(I apologize if this has been posted before, but I searched the forums and didn't get any results on this topic.)
There's a Save option in the Character Designer, but it seems to be unique to each character and, as far as I can tell, the data doesn't get saved anywhere where it might be shared with another player.
There are jpg’s in your screenshots folders that have your character designs. I’ve not tested it but I’d not be surprised if that would give somebody your character design. The only thing to remember is what your buddy makes has to be the same species as what you made or it won’t show up as an option. Also don’t use special outfits for the save that you’re going to give out. I’ve notice even between saves on the same computer and account if I save one file with a skirt in it the save will not open right on other toons.