Clearly, in order to implement things like Lead-Lease or the partition of Iran, there had to be a certain amount of back-and-forth contact between the Soviets and the Western Allies. I would not be surprised at all to find out about American or British officers serving at their posts inside the Soviet Union. But I assume that your question is specifically about combat units, and actual fighting.

I am aware of only two incidents, both in the air forces.

The (British) RAF stationed part of the 151st Wing in Murmansk for a short time, mostly to help train Soviet pilots on the British aircraft they were being given. But they did go into combat, and score kills against the Germans (and suffer one combat fatality).

Here is a Daily Mail newspaper article about this unit: http://www.dailymail.co.uk/news/article-1181390/Stalins-British-heroes-The-discovery-forgotten-medal-reveals-extraordinary-courage-RAF-aces-fought-Soviet-Union.html

The other incident is the Free French "Normandy" squadron, later called the "Normandie-Niemen". These guys fought alongside the Soviets from 1943 to the end of the war. Stalin gave them a bunch of medals, as well.

Here's an article from the Smithsonian Air and Space magazine about them: http://www.airspacemag.com/military-aviation/the-french-russian-connection-8043795/?no-ist

This unit still exists as well, as part of the French air force. They have a web page (en francais) with unit histories here: http://rc230-normandieniemen.com/

My uncle (US) was flight engineer on a B24, and got shot down over the eastern front, and spent 28 days as a German POW. So yes, not sure of many more details,
But at least US had air corps in SE Europe. They had been warned if forced down,
aim for the German side, as the Russians had no food and wouldn't help or repatriate
allied air crews.