Hi,
Since this Thread last had a post it has been found the most likely candidate is an issue with the Apple SNATMAP server.
The issue effects both Jabber and AIM connections for Video and Audio chat and Screen Sharing and from at least iChat 5.0.3 in Snow Leopard (I also suspect earlier versions).
The only place on the internet these all come together is at the SNATMAP server.
It seems that only non NAT connection to the Internet work. (Such as Setting an Apple Base Station to allow the Mac to do the Login to the ISP)
Some other routers allow this but in some cases people have routers with the Login detail hard coded into them and they cannot do this.
It is also impractical for some people running two or three computers (Not to mention iPhone, iPads and satellite boxes that use the Internet for On demand stuff plus Apple TVs etc.).
As you know the Messages app will invoke FaceTime fro Audio Chats and Video Chats from the Chat window to iMessages contacts (Your need to be in a conversation with them).
In Yosemite to other Yosemite computers it will also do Screen Sharing by Invoking the Screen Sharing app (You have to allow it in System Preferences > Sharing).
As you may realise this will not be an answer for those still using earlier version of iChat or Messages. (it's a work around not a "Fix")

8:14 p.m. Friday; September 25, 2015
iMac 2.5Ghz i5 2011 (Mavericks 10.9)
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