Here is the Google Chrome Web Speech API demo page: https://www.google.com/intl/en/chrome/demos/speech.html
Google Chrome does use a cloud service.
Edit: Firefox does not support the Web Speech API at this time. There are not currently any offline versions of this API as far as I can tell.
https://developer.mozilla.org/en-US/docs/Web/API/Web_Speech_...
EDIT: According to other documentation, it is behind a config flag. If you enable it, it will send the data to Google's API through a Mozilla-operated proxy: https://wiki.mozilla.org/index.php?title=Web_Speech_API_-_Sp...
It looks like there is no offline version :(
Here is the Google Chrome Web Speech API demo page: https://www.google.com/intl/en/chrome/demos/speech.html