I don't think there's anything out there to fit your needs out of the box.
I think that, in theory, you could grab a screenshot of your Stage every x milliseconds and send it to the other end. But that would be a huge amount of data to transfer. You should work out an algorithm that sends only information regarding what pixels have actually changed in your screen to make it more efficient. Maybe you could port an Open Source C library?
Once you have a super-efficient algorithm, you could use the new P2P capabilities of Flash player 10 to speed up the interchange of information. But it looks like an awful lot of work.
You could also go the Open Source route, forget AIR and its limitations to launch external apps and use SWHX together with any C library to do the actual job. Sounds much more plausible to me.
Or maybe there's something done already that I'm just not aware of.
I leave this more as a bonus really, Darron Schall wrote a VNC client in AS3 time ago, it's not exactly what you need, but it's related.
Good luck : )
Juan