In October 1943 Mp43s were sent to HGr B (deployed in Italy at the time) and thus to the units of 71.Inf.Div.,GrenDiv “Hoch u. Deutschmeister” and LSSAH. I have no number of how many were issued but I would say few.
After that issue im not sure whether more were sent to Italy but I doubt it. Most MP43/44 were sent to the Eastern front to Heer and some Waffen-SS units. Also many were issued to the new Volksgrendiere divisions to add to their firepower.
I have seen partisans armed with Mp44 as late as 1944 so its not for certain if more were sent there. Also I have seen a photo of an FJ with a Stg44 but I have no clue if that one is from Italy or Normandy or perhaps the eastern front. Ill try to find the pics on my harddrive when I get back from work!
ource - Sturmgewehr! From Firepower to Striking Power, Hans Dietrich Handrich, Collector Grade Publicatio.