If you’re using Chrome, you can navigate to chrome://webrtc-internals. This will show you the offer, answer, ICE states, and statistics about the connection (once it has been established).
For more in-depth debugging, you can see logs of all the STUN pings between candidates by starting Chrome with the following flags:
--enable-logging --v=4
The logs will be in the chrome_debug.log file in your user data directory: http://dev.chromium.org/user-experience/user-data-directory