The autopilot panel and servos plug into the Skyview network so they would not be affected.
The transponder must be connected to the same serial port on both screens to function if one screen goes down. For instance, if connected to only the left screen and it is powered off, you would have no control over the transponder.
The AP may disconnect and the transponder may go offline for a moment, but if that happens both will come back automatically within a few seconds. The SkyView system is designed for complete redundancy with two screens.