Do left and right mic buttons control both headset microphones? The aircraft I was flying today has a push to talk integrated button on each control yoke. I was sitting in the right seat, but my headset was plugged into the jacks on the left side of the cockpit. When I pushed the right mic button, the tower could hear a carrier wave but no voice transmission.  It was only when I pushed the left mic button that they could hear my transmission. Is this normal?  Seems like pushing either mic button should allow a transmission from either headset.
This is the expected behaviour. The other pilot might make some sound or even talk to someone else. You don't want any noise interfering with your own voice during transmission.  Two microphones picking up your voice might also decrease the sound quality because of the small delay between the two locations.