I have a logitech webcam. The Shield pics up the mic and video no problem. The only issue is hangouts produces no sound at all. It doesn't make the ringtone noise and when someone picks up it's silent. They can hear me through the mic and they can see my video. I downloaded a volume control app to see if the call volume was at 0 and it was 4/5 so that doesn't seem to be it either. I'm not sure what else it could be.
Ask on XDA , they probably be able to help better