Does Facetime on Mountain Lion allow you to share your screen? I am trying to figure out a way to share my screen with someone else.  I wanted to see if Facetime had that feature or if there is another recommendation for screen sharing on Mountain Lion.
There is not a way to do this yet in FaceTime.  Here's the link to a forum where they discuss how "neat" it would be to have this feature.  The discussion is a month old, so fairly recent.

As for Screen sharing, if you're on the same network (either routed, through a VPN, SSH tunnel, or direct connection) then you just select the device in Finder, and choose screen sharing (button in the upper right hand corner) in the window that opens:



The computer that is sharing its screen must have Screen sharing turned on (check marked) in System Settings > Sharing:



Note mine is NOT check marked, so this computer would NOT share its screen using screen sharing.

If you want to Screen share and you are NOT in the same Network, use Messages:



Note Screen Sharing (under the video drop down)is enabled (has a check mark next to it).