I've read estimates that production of the MP44/StG44 was as much as 5000 a month and as mentioned by Panther Al, it started to be seen in western Europe particularly during the time of the Ardennes campaign. I think however, that by the time they appeared in enough numbers on the Western Front, it was "too little, too late" for Nazi Germany and so the impact wasn't going to be anywhere near as significant as it was on the Eastern Front.
And speaking of the Ardennes campaign, enough StG44's were available on the Western Front that some of them showed up in Spain and were available for filming of the movie Battle Of The Bulge in 1965, (filmed in and around the Sierra de Guadarrama mountains near Madrid).
Internet Movie Firearms Database page for
Battle Of The Bulge
There were also enough of them available in the US during the 1970s/1980s for them to appear, highly modified, as weapons of the Rebel Alliance on the planet Hoth in The Empire Strikes Back movie.
The StG44 may even have made an appearance in the Western Desert with the DAK, there's one on display in the El Alamein War Museum (although knowing how little effort is actually made by the Egyptians to research WW2, it's also possible that the example on display came from a post-WW2 North African conflict).