score:10
The German Wikipedia article lists the following numbers (most likely taken from Salvador Bofarull's book "Pigeon mail through history" which I couldn't find online, the numbers are confirmed in a bunch of other places however):
I couldn't find any numbers for the Soviet army, merely the continuously repeated statement of unclear origin mentioning 15,000 delivered messages. Compared to the numbers above this would mean that the Soviet army barely used homing pigeons. I also couldn't find evidence for any significant use of homing pigeons by the German army in WW2.
You can see how the pigeons were used by looking at the "celebrities":
Btw, it seems that the use of falcons by Germans is not confirmed and likely didn't happen. Pigeons were simply shot, both in WWI and WW2. Also, the "Brieftaubenverordnung" of 20th September 1940 banned homing pigeons in the occupied territories of Belgium and France.