No. Stack Exchange is probably never (ever, see meta.se) going to have a private messaging system. Private communication methods exist for official moderator business, but that's about it.
Remember, we're first and foremost a Q&A network that's here to build a public wiki to help people through the problems they're facing, and while our chat's a great medium to play games on, it's there to support a Q&A network, not to support RPG gameplay. If you want to be able to share notes, I suggest you exchange instant messenger details with the other players or something.
In light of our wiki goal, having PMs opens us up to one of the more counterproductive acts available, which is people using them for Q&A. (This happens too often on forums.) Q&A happens here in the public eye where (a) everyone sees the author's clarifications and updates, (b) everything gets peer reviewed (and terrible ideas like "reinstall your operating system" get called out as such), (c) you can see things everyone's suggested so far, and (d) when it's finally solved, it's solved in public where everyone can access the solution forever.
None of that really happens in PMs, so they don't help our goals and probably undermine them. (I also left out all the potential scam/harassment/exploitation scenarios.)