I'm American, and I was thinking about how even though Russia is our "enemy" for many decades now, I don't think a US soldier and a Russian soldier have ever tried to kill each other while in uniform. But I could be wrong.
Well the most recent example I could think of off the top of my head would be the air battles over "Mig Alley" in Korea. The Soviets provided advanced MIG-15 fighters to the KPA, which gave them an advantage over American fighters then deployed to Korea. However, the Soviets provided much more than fighters and equipment. They deployed hundreds of pilots and ground crews to fly and service these MIGs. Over 300 Soviet built MIGs were destroyed during the Korean War, and over 100 Russian pilots were killed. Im not sure how many kills Russians (as opposed to Koreans) claimed, but I know they minted several fighter aces in the battles over Korea.
Following World War One, elements of an American Infantry Division were deployed to northern Russia in an attempt to remove western military hardware given to the Russian Empire, but now threaten by Bolshevik forces. The so-called "Polar Bear Expedition" invovled several thousand American troops who were supposed to be used only as guards for the materiel. However, because the weapons had already been transferred south, the American forces were deployed in offensives actions with the aim of rescuing that material. In the process, between 200-500 American soldiers became casualties as a result of enemy action and disease.
There are likely other examples of Russian and American casualties. During the Second World War, for example, thousands of ethnic Russians were deployed to the French coast to bollster the forces defending against American attack. These forces, then serving in the German Army, came under intense attack during the first days of the Normandy invasion. However, I think this question is really asking about soldiers fighting for the Russian Military and the American military respectively. I think the above examples are probably the largest examples of that kind of conflict in the 20th century, but not the only ones.
In terms of direct military conflict with casualties, one example comes to mind. On November 7, 1944, there was a little-known air battle over the town of Nis, in Serbia. A column of moving Soviet vehicles of the 6th Guards Rifle Corps was strafed and bombed by P-38 Lightnings of an unknown unit (most likely 15th Air Force, so either the 1st, 14th, or 82nd Fighter Groups), killing 31 Soviet soldiers and wounding 37. Soviet Yak fighter planes then arrived from a nearby airfield and began to fight the Lightnings. It gets quite sketchy here as many documents are still classified, but anywhere from 2-4 Lightnings and 3-4 Yaks were lost.
We know that the Soviets lost at least three planes, with at least two pilots killed.
"The losses (of the) 866th Regiment (were); "Lightning" shot down two Yak-9. One Soviet pilot was burned along with the aircraft, the second was seriously injured and was sent to the hospital. Another of our fighter, pursuing the American, got into a zone of fire of its airfield air defense and was shot down. The pilot and the plane burned. From his injuries he died in hospital and the commander of Infantry Division, General Stepanov - the father of the pilot Victor Stepanov."
The Americans also lost at least three P-38s, with at least two pilots killed
Incurred losses impressive. Here is what an archival document (says); As a result of defensive air combat our fighters shot down three aircraft "Lightning." One plane crashed a mile northwest of Nis, the plane and the pilot were burned. Another fell a mile northeast of the airport, the plane burned, Pilot died. Both bodies were buried in the vicinity of the airport ... The third fell eight kilometers north of the airport, in the mountains. The aircraft, surveillance from the air, burned down completely. Two other aircraft (P-38), lined (hit) in a dogfight, with smoking engines left at low altitude in the south-west. (On) One of the dead pilots (was) found an identity card and a map of Italy without the route and (or) any notes ... "
and
"(The) Air battle flared up even more. The Americans are dropping bombs, first tried to defend himself. But, unable to withstand the onslaught of our fighters, built in the "snake" to better cover each other with fire (from) front machine guns, and went out toward the city. One of the "Yakov" promptly dived from a height on (enemy) aircraft and opened fire. 37 mm projectile (from) his gun exploded in the center section "Lightning" and he blushed like a torch, (and) fell to the ground."
Due to the circumstances of the event, it is entirely possible that the Americans had strayed far off-course and mistook their location for northern Italy, and the Soviets for Germans.
Sources:
Article about the event (Translated from Russian so excuse the grammar; I have cleaned it up a bit)
P-38 Lightning in Action, by Gene B. Stafford
Would the Russian revolution/civil war count?