I'm trying to chase down information on this claim. Wikipedia seems to be pulling that statistic from James Dunhill's 2000 book with the click-baity title Dirty Little Secrets of the Vietnam War: Military Information You're Not Supposed to Know. Regarding those casualties Dunhill states:

Apparently an annual contingent of about three thousand Soviet troops served in Vietnam from about 1965 to the fall of Saigon. By some accounts, it was Soviet antiaircraft personnel who shot down the first American aircraft lost of North Vietnam in 1965, but thereafter the Russians restricted their activities to serving as technical advisers. Although Russian sources state that no Soviet personnel served in South Vietnam, on at least one occasion there was a report of a uniformed white man serving as a a consultant to an NVA commander in the south. According to official Russian sources, 16 Soviet military personnel were killed during the Vietnam War.

Unfortunately, Dunhill has no footnotes or bibliography for his book - a quick perusal of his background seems to show that he's a wargame designer who put out a whole range of "Dirty Little Secrets" books, which is all well and good, but is less than ideal as an historical source. Anyway, chasing down what these "official Russian sources" seem to be led me to Soviet Casualties and Combat Losses in the Twentieth Century, by Grigorii Krivosheev, the English version of which was published in 1997. The most it says about Vietnam is mentioning it in a table in a section on a larger discussion of Soviet military personnel who were deployed, often in extremely high secrecy, to friendly countries in the Third World in conflict, such as Vietnam and Egypt during the Continuation War. These troops usually served as technical advisers, engineers and anti-aircraft troops.

It's worth pointing out that the People's Republic of China performed a similar role in North Vietnam during the war, but in much higher numbers, with tens of thousands of Chinese troops manning anti-aircraft batteries in the North, and performing engineering and logistics work along the North Vietnam - Chinese border. In the Soviet case this was part of a massive aid effort, which involved the Soviet Union providing aid money and massive shipments of weapons and materiel to North Vietnam to keep up its war effort. A short description of that relationship can be found here.

Which is all to say, these Soviet casualties were most likely from among a group of deployed troops acting as advisers and technical personnel. But the only mention of any of them being in South Vietnam even in this capacity seems to be from an unsourced second-hand rumor. It seems like "Russian/Soviet Soldiers" was almost something like a Bigfoot Sighting phenomenon among American forces in Indochina - it was something people talked about (or claimed to have heard Russian on radio frequencies), but no one ever had conclusive evidence - definitely not a prisoner or a body. Unlike in the Korean War, there doesn't seem to be any evidence of Soviet personnel and US personnel engaging each other in combat.