If you find yourself trying to use your FaceTime Camera in a browser (like for a Google Hangout) and your camera can’t be found, try killing the AppleCameraAssistant process. It’s easiest to do this via the Terminal app with tho following line:
sudo killall AppleCameraAssistant
Then try to turn on video in your hangout, and it should work straight away, without even needing to rejoin the call.
I’ve had issues with Safari and Chrome not finding my camera since the very first Mavericks beta. My hunch as to why this is necessary, is Skype having a tendency to reserve the camera process for itself, and browsers as secondary citizens can’t kill it themselves anymore.
Oddly, native OS X apps that rely on the FaceTime Camera (FaceTime, Photo Booth, etc…) have no issues — it’s just a browser quirk, but killing the process resolves it.
Hope this helps!