A:Answer The way the headset works, if configured properly is that it creates two audio outputs for your system. In one you'll have your game and OS chimes and whatever else you want while on the other you should have only the chat voice audio out. The headset put them both together for you in the end, allowing you to control the volume between them. Simple technique applied cleverly for a great result.
However, when recording, most recording solutions will only take ONE audio input. That is why it recorded the game (audio out 1) and your voice (mic), but your friend's voice (audio out 2) was not recorded.
A simple solution is to configure your chat applications to use the same audio out (game) as the game and OS. You'll loose control over voice versus game volume, but your videos will record chat audio as well.