I can answer a small part of the question about France.

I assume your question is for the allied side, because they were french soldiers&volonteers on both side on the eastern front.

The free french (under command of General de Gaulle), based in london since the fall of France sent a fighter pilots wing to fight soviet side on the east front.

It was the Normandie-Niemen Regiment.

The first objective was for public relation purpose, Free French were eager to show the world that french were still fighting.

They were fully equiped with russian equipment (Yak) (and supported by russian technical staff).

But the pilots were experienced and trained french pilots, often from regular french armies that defected to london after the battle of France. The regiment performed quite good.

It lots around 50 pilots during the war, but 30 pilots reached ace status and claimed more than two hundreds kills.

They had tons of french and soviets medals and there are still ceremonies today about this regiment both in France and in Russia.

The Germans had their Ostruppen -- people from the soviet union who joined the german army. Some were recruited from occupied territories, others from POW camps. These were 2nd and 3rd line units typically used for policing operations and to man fixed fortifications in quiet sectors. Like beaches in Normandy when the attack was expected in Calais. Specifically the 243rd and 709th static infantry divisions who were stationed near Gold, Juno and Sword beaches.

So, soviet soldiers fought against the western allies under the German flag.

To add to the other answers, some Soviet soldiers found themselves in Resistance groups in France. In Burgundy, the Leningrad Detachment left some memories in the population. Both good and bad ones due to their violence toward collaborationists ( or suspected ones ) and several acts of thefts but also because they participated in resisting actions.

These Soviets were Red Army P.O.W. who had escaped German labor camps in France. When Soviet soldiers escaped they were generally taken care of by the FTP-MOI ( Franc-Tireurs Partisans-Main d'Oeuvre Émigrée ) which was in contact with Moscow via the French Communist Party. There little sources about the Leningrad Detachment. Gérard Soufflet, a local amateur historian, with the help of Jean-Marc Berlière a professor at the University of Burgundy, wrote a book about the detachment. Most of the sources are from the French police ( investigating the presence of a " Russian working with the Polish terrorists " ), post-war Polish Intelligence and also " local " memories and photographs.

The Leningrad Detachment is not the only one. Most of the Soviet who escaped prisons and labor camps were sent to zone where a lot of Polish participated in the Resistance due to the relative similarities between culture and language. The Saône-et-Loire ( in Burgundy ) mining region had a lot of Polish migrants living and working there ( perhaps around 30,000 ), many were Communists.