In a 2-Rx by 2-Tx directional antenna system, say both transmitters are transmitting concurrently as in Fig. The main lobes from the two transmitters overlap. How does this affect the resultant signal received at either receivers ? Any thoughts on this would be greatly appreciated !
Free space is (under normal conditions of frequency and power and atmosphere) a linear medium, so waves propagating in it follow the superposition principle, so they do not interact just because they occupy the same space. The situation in your diagram is not particularly significant.
What you do need to consider is the rest of the radiation pattern of your antennas, specifically the part of TX 1's pattern in the direction of RX 2 and the part of TX 2's pattern in the direction of RX 1. These will have an effect — hopefully a negligible one.
The fact that the beams cross does not matter; the exact same analysis would apply if they were parallel instead.