I busted out wireshark to answer my own question. The data is actually getting encoded as speex and being posted to http://www.google.com/speech-api/v1/recognize . Maybe google is about to open up its speech recognition API to the masses?
If it's anything like speech recognition in Android, it's all server-side..so this should be technically possible in any browser as long as google allows it.
You can't see it in the Chrome Developer Tools/Network UI, but the browser is sending the recorded audio input to a webserver. Unfortunately it's HTTPS, so a bit hard to decode the URL/content of the message without using a proxy.