Hi,
AS has been said the Messages app in Mountain Lion can "do" iMessages (the same sort of IOS Messages that iPhones and iPads can do).
As you say, this "side" cannot do Screen Sharing.
It can also use Buddy Lists (AIM, Jabber or Yahoo Logins plus Bonjour).
These Buddy Lists are not on Screen in the default Set up (See Window Menu > Buddies) but when they are on show they are a Combined List (Now See the General Section of the Preferences to separate them).
Any Yahoo Account cannot do Video and Audio Chats and therefore no Screen Sharing.
Jabber (Facebook, GoogleTalk IDs amongst others) can Video and Audio Chats iChat to iChat (or Messages at either end) so can do Screen Sharing.
AIm Logins (@Mac.com, @me.com as well as AIM registered names can Video to the AIM for PC app and Trilliian (with Pro Add on) on a PC as well as iChat/Messages combinations.
Screen Sharing is Mac to Mac with either Messages or iChat.
IF you have separate Jabber or AIM Buddy lists on show then bring one to the "Front"
The Video Menu > Screen Sharing should now be dark and you will be able to turn the Sharing On or Off (Selecting it toggles it to the opposite)
IF the Screen Sharing items is greyed out then it tends to reflect an issue with Internet Speed.
When iChat or Messages Starts up it checks on the Internet Speed that can be seen.
This, coupled with the setting in the Preferences > Video Section > Bandwidth drop down, provide a stored figure in the .plist
This figure has to be at least 128kbps both as upload as well as download.
Obviously it applies to both ends.
If Messages or iChat start up when the Computer does then the figure the app "sees" can be distorted as the Computer establishes an Internet Connection.
Screen Sharing is essentially an Audio Chat+
The Plus bit is the VNC connection that is made to Display the Screen/Desktop.
This VNC connection takes place on a random port so the routing device has to be using UPnP or Port Mapping Protocol if it is an Apple Base Station of any sort.
However this will not grey out the line in the Video menu.
So...
No the iMessages side will not Screen Share.
If you have not got another account set up you will need to do so.
It will need to be the same type as your girlfriend (Both AIM or Both Jabber).
8:06 PM Friday; November 9, 2012
Please, if posting Logs, do not post any Log info after the line "Binary Images for iChat"
iMac 2.5Ghz 5i 2011 (Mountain Lion 10.8.2)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad
"Limit the Logs to the Bits above Binary Images." No, Seriously