The radiated signal decreases by the square of the distance. An omnidirectional antenna can't focus the energy it has available in the desired direction so the power going in any specific direction is lower. Then distance comes in.
The transmitted signal (from the perspective of the AP with a directional antenna) also decreases by the distance squared. And since the distance is exactly the same in both directions, the transmitted signal weakens by the exact same ratio as the received signal. So distance cancels out when comparing transmitted vs received signal strength.
If you're using the same antenna for send and receive, then both signals go through the same transformation between circuitry and RF in the air, so the gain (signal amplification) is the same for send and receive.
Say you have two identical omnidirectional wireless APs. You place them a distance x apart.
- AP_A transmits with strength S. It gets received by AP_B with strength S/x^2.
- Likewise, AP_B transmits with strength S, and it gets received by AP_A with strength S/x^2.
Now say you move them apart to distance 10x, and replace the antenna on AP_A with a directional antenna which boosts signal strength 100x.
- AP_A transmits with strength S, which the antenna boosts to 100S. This gets attenuated by distance to AP_B to 1/(10x)^2 = 1/100x^2. The signal AP_B sees is thus strength 100S(1/100x^2) = 100S/100x^2 = S/x^2
- AP_B transmits with strength S, which gets attenuated by distance tp AP_B to S/100x^2. But the antenna at on AP_A boosts received signal strength 100x, to 100(S/100x^2) = S/x^2.
And so putting a directional antenna on only one end results in the same signal strength in both directions.