How do pilot and copilot operate and coordinate the radio?
Both yokes have PTT buttons. Generally one person is in charge of the communications. In military aircraft that is often the task of the navigator in twin crew airplanes. I would think it generally falls on the pilot not flying the aircraft although both should be listening to the instructions. 

The switch is on the front of the yoke (reverse side from what the pilot sees) and is generally triggered with the left index finger. Here is a pic from a yoke for reference. Note the button on the left upper part labeled "mic".


source

I have never seen it (but then again there are many radios I have not seen) but some radios may have a VOX (voice-operated transmit) option which would bypass the PTT. This however would be very irritating on the channels since there are lots of planes on a single frequency talking to the tower.