What are the relative efficiencies of the various stages of a counter rotating propeller?
The key thing to think about is that the first stage doesn't just accelerate air downwards, it also causes it to rotate ("swirl") in the same direction as the first stage propeller.

This happens because the first stage propeller, being an airfoil, generates both lift and drag. The lift corresponds to air being accelerated down the axis of the duct; the drag corresponds to air being accelerated circularly around it. The amount of swirl force is therefore related to the thrust and the effective L/D ratio of the propeller.

The airfoil of the second stage propeller is then encountering air that is moving towards it at the tangential velocity of the propeller, plus the tangential velocity of the air that has now been circularly accelerated ("pre-swirled") in the opposite direction by the first stage.

That increased velocity means that the angle of attack for the second stage airfoil will be increased compared to that encountered by the first stage, and hence greater lift is produced at the same RPM.

To balance the load on the two stages you would need to adjust the pitch or RPM of the second stage propeller.

Note that after the second stage, the air will be "unswirled" back to being closer to a non-rotating flow, and this is one of the reasons that a contra-rotating pair can be more efficient than a single propeller: overall, the system wastes less power in swirling the air rather than accelerating it axially.