If the second player is a silver account, the online chat feature will turn off as that account seems to take precedence when playing online. If you want to chat while playing with a silver account player, you will have to use a headset.
This was a suggested answer off of the Xbox forums to a perhaps identical problem.
Outside of that, the Microsoft help suggests the following generic checks, though this likely has nothing to do with having the second player connected:
From the Xbox 360 Guide
1. On your console, sign into Xbox LIVE.
2. On your controller, press the Guide button.
3. Select Chat & IM.
4. Select Kinect Microphone to turn the microphone on or off.
For more basic Xbox LIVE Chat troubleshooting, check here: http://support.xbox.com/en-US/xbox-live/connecting/parties-voice-chat