As with the other comments, the Soviets never trusted German POWs enough to allow them their own formations. There was an organization called the National Committee for a Free Germany consisting of emigres and POWs that hewed towards the Soviet cause. Most of their efforts were in psychological warfare and propaganda. For example, Friedrich Paulus joined after the July 20 Plot.

One of the more tragic notes of the Great Patriotic War was that the USSR still had many Volksdeutsche populations. In the Volga basin, they had actually managed to form their own ASSR in the interwar period of around 600000. In the beginning, the Soviet state averred that this was a war against fascism, not the German nation. In a June 22 speech, Molotov asserted that "this war is not forced on us by the German people...but from a clique of bloodthirsty fascist rulers in Germany." One of the first heroes of the war was a Volga German, Heinrich Hoffmann and the Red Army newspaper published an account of his fatal defense along with pictures of his bloody Komsomol book. This international approach was short-lived and soon being ethnically German became incompatible with being a Soviet citizen. The state banished the entirety of its German population (about 1-2 million) into harsh work camps in the interior of the USSR. After the war, they were expelled out of the USSR into the Eastern bloc.

sources

Berkhoff, Karel C. Motherland in Danger: Soviet Propaganda During World War II. Cambridge, Massachusetts: Harvard University Press, 2012.

Brandenberger, David. National Bolshevism: Stalinist Mass Culture and the Formation of Modern Russian National Identity, 1931-1956. Cambridge, MA: Harvard University Press, 2002.

Schlögel, Karl. Russian-German Special Relations in the Twentieth Century A Closed Chapter? Oxford: Berg, 2006.

From what I've researched too, I have never read about or heard of any German units fighting for Soviet Russia. There are a couple likely reasons for this:

-not a whole lot of Germans lived in the USSR. Most Germans that lived outside of Germany (before WWII) lived in western and northern Poland, Austria, Czechoslovakia, Switzerland, and other Central European countries.

-Stalin, being the wily, paranoid character that he was, would have certainly not liked the idea of a large German unit learning the Red Army's ways. Perfect opportunity for spies/double-agents for Germany. The notoriously bad treatment of Germans by the Soviets also would be a huge deterrent.

-Germans were indoctrinated more or less to despise communism, especially the Soviets. After having a hugely ineffective government after WWI largely run by communists/socialist, along with a bloody rebellion immediately after the war that saw revolutionary socialists fighting their opponents in the streets, Germans weren't so big on the idea of communism.

There was Czech corps (three infantry brigades, tank regiment and artillery regiment) and two Romanian infantry divisions in the Soviet army by the end of 1944, but German POW's never been trusted enough to form their units in Soviet army. Although apparently they've been used in hiwi-like way (truck and car drivers mostly).