This is quite a problem! I'm on my laptop, with a built-in microphone, so what I've resorted to doing when I don't want people to hear me or what's in my background, I disable the microphone under Recording Devices. It's not push-to-talk, but it saves others the trouble of muting me or listening into private conversations.
There's no "Push-to-talk" option to speak of, that I can find, yet. They should make it more obvious, or add one in.
