Signal return path between circuits with separate full-bridge rectifiers
Unless your two circuits, the TX block and RX block are galvanically isolated from each other you need to have a ground (signal) return path between the two circuits.  The currents that flow into your RX device (and there will be some, even if they are small) need to be able to find their way, so to speak, back to the TX block and its power supply.  Remember that current always flows in a loop. 

There may be a sneak return path back through the blocks labeled BR, but without knowing the details of what's in those blocks it's hard to say.

Even if such a path exists, it may suffice for DC purposes, but may not work for high speed, even moderately high speed, signals.

EDIT 1 - Added current flow graphic