How do the Beech Bonanza's ruddervators work?
In the Bonanza, a mechanical mixer controls ruddervator movement based on pitch and yaw commands. Here's a typical V-tail mixer (this is not of a Bonanza though):

(source: eaa1000.av.org)
There are two rods extending off to the left of this photo that connect to the actual ruddervators. Rightward motion of the top rod (due to either rightward motion of the entire mixer assembly due to a pitch command, or clockwise rotation of the assembly due to a yaw command) will deflect the ruddervator one direction; leftward motion deflects it the other direction. Similar for the other ruddervator connected at the bottom.
I can't completely answer the second part of your question, whether yaw authority is reduced when pitching or vice versa. It depends on where the limit stops are installed in a Bonanza. If the stops are set on the ruddervator side (left side in photo above, so that the mixer's range of motion is limited by its leftward connections), the amount of available rudder will be reduced when pitching. (Sorry I can't answer that part of the question, but I thought the information about the mixer would still make a useful answer.)
The airplane is obviously designed so that sufficient authority is available in all situations. (This means making the two fins of the V longer than those of a traditional horizontal/vertical stabilizer design so that similar control surface area is available.)