Its not the headset itself that does this for the most part, but rather the voice service being used.
Teamspeak is one such example, you can set a noise gate on it so once a certain volume is reached it begins to transmit and stops when it falls under it.
This can be used to avoid background noises that may be around.
However it is not a flawless system. Most of the time the first portion of what is said gets cut off, activating the gate, and if you fall below the level you stop transmitting.
Overall, push to talk is good to use. That or always transmit if you can be silent when not voicing.
Other than that, there really arent any headsets like you ask.