A yes and no answer.
The app has no abilities on this front itself.
However Quicktime Player can be used to record the screen during a chat.
The FaceTime App has to remain in front of others if you do this.
There is also Screeny
iMac 2.5Ghz 5i 2011 (Mountain Lion 10.8.4)
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
I think what everyone is missing in the original question is the value of recording Facetime Chats in the context of easy to record Video Podcasts. This is a great concept for "Interview" style Q&A, "How to training" and similar positive utilizations. There are Display Recorder apps that worked prior to iOS 6.1.3. This is a great concept and ought to be explored