Yes, of course. Not only the total number of PPSh-41 (not to mention the PPS-43) produced is way higher than the MP-44's, but the percentage of PPSh-41 used in the Soviet forces was probably higher than the percentage of MP-44 used in the Wehrmacht.
At least partly because, Hitler not being exactly the most supportive for a project about a weapon he maybe considered redundant with what was already in use - MP-40 and later models ; plus other machine pistols, less known and used in lesser numbers, but still present in the German soldiers hands - in the German forces, the MP-44 was confined for quite a time to the state of quasi-secret prototype. The engineers responsible of this even disguised the MP-44 as Stg-44 (Stg = Sturmgewehr. Literally, “assault rifle”), or as "automatic carbine model 1944".
Now, concerning the ammo, of course, given the numbers of PPSh-41 and PPS-43 - which fired the same ammo : 7,62 x 25 mm TT, or 7,62 Tokarev - used by the Soviet soldiers, the ammo was abundant, very abundant. The problem was the actual possibility, for German soldiers, to get it from their unwilling Soviet colleagues. Though I once read that, in their initial advance in Russia, the Germans were able to seize great weapons and ammo factories, along with very great quantities of 7,62 Tokarev ammo.)
But both weapons had their inherent qualities and flaws. The PPSh was incredibly resistant to misfire and jamming, and sported a whopping 71-bullets drum magazine, while the MP-44 was probably more accurate and had a better ergonomy. And inspired the AK-47 to Mikhaïl Kalashnikov.