The bulk of combat operations by the communist side in Korea were taken up by the Chinese from October 1950. The Chinese forces, fighting under the name "People's Volunteer Army", were regular People's Liberation Army units sent to Korea in order to check the UN advance north of the 38th Parallel and protect Chinese interests by maintaining North Korea as a buffer state against western-aligned powers.

This intervention was extensive, and resulted in several major offensives between October 1950 and summer 1951, at which point the front line stagnated and operations devolved into general trench warfare-style combat with small-scale offensive operations pursuing limited goals. It was actually a major detour by the Chinese, who had been planning to invade Taiwan to destroy what remained of Nationalist China, and had to shift many forces to Manchuria instead.

Chinese activity in Korea was dominated by the use of aggressive infantry tactics with limited artillery and virtually no armour support. The UN held air superiority during the entire conflict, which left the Chinese dangerously exposed, and their commander, Peng Dehuai, regularly complained to his superiors in Beijing that his men were sitting ducks and he needed air support. Initially this came in the form of Soviet anti-air units, and air units equipped with MiG-15s (in Chinese uniforms and paint schemes - they were told to use Chinese on the radio as well, but few ever did), and eventually Stalin relented and allowed Chinese and North Korean units to also fly these highly advanced jets. Still, the communist fighter air defences rarely left an area around the far northwest of Korea known as 'MiG Alley', for fear that a downed pilot may be lost as a PoW or a downed plane may be salvaged by the UN (it was top secret). In addition, the Chinese pilots suffered heavy losses against UN pilots - Chinese desperation only allowed for a limited number of hours in training flights, fewer than 100 per man. The average for US pilots in particular, if memory serves, was something in the order of 300-600 hours, and many of the pilots over Korea were Second World War veterans, as were many of the Soviet pilots in the intervention.

Back to on the ground, as I mentioned, the Chinese took up the bulk of the war effort. By October 1950, a huge part of North Korea was in UN hands, and the Korean People's Army had been mostly routed or destroyed, what remained hiding in the frigid mountains of northern and northeastern North Korea. At the end of the month, at Unsan, the Chinese riposte saw two South Korean divisions wiped out, and over the next few days significant portions of the US 8th Cavalry Regiment were also badly mauled or eliminated. The intervening Chinese force as of November was estimated by the UN as between 180,000-380,000 men, in approximately 30 infantry divisions and a number of supporting units. Compared to the KPA, which had initially numbered between 150,000-200,000 at its height in the early days of the war, this was significant, and would only grow as the Chinese offensives pushed further south - approximately 40 infantry divisions by July 1951.

This numerical superiority lasted until the end of the war, and the Chinese would remain in direct contact with American units for that whole period. It was the Chinese who pushed the entire UN line back beyond the 38th Parallel and recaptured Seoul for the communists in January 1951. It was also the Chinese who failed to strike a decisive blow with their 'Fifth Phase Offensive' in April 1951 due to a rallied UN force now commanded by General Ridgway and successful holding actions at the Imjin River and Kapyong. Once the line had stabilised, much of the communist fortification network at the front was manned by Chinese units, with the North Koreans having a token presence towards the eastern end, and much of their army in reserve as it had to almost be rebuilt from scratch.

Just a quick note before I finish up regarding the nature of the Chinese fighting specifically. The Chinese soldier was considered a formidable opponent - during the actions of November-December 1950, around the Chongchon River and Chosin Reservoir, they showed that they could maintain offensive actions with no armour or air support in bitter cold weather on a fraction of the supplies required by their American opponents. They could be very aggressive in action, emphasising close combat and infiltration to nullify the UN advantages in artillery and air power. This meant that the Chinese and Americans got up close and personal on a very regular basis from winter 1950 onwards.

Sources:
Kevin Mahoney, Formidable Enemies (Novato CA, 2001)
Allan R. Millett, The War for Korea, 1950-51 (Lawrence KS, 2010)
Gordon L. Rottmann, Korean War Order of Battle (Westport CT, 2002)
Nigel Thomas & Peter Abbott, The Korean War 1950-53 (Oxford, 1986)
Xiaoming Zhang, Red Wings over the Yalu (College Station TX, 2002)

Between June 1965 and March 1968, there were approximately 320,000 PLA soldiers in North Vietnam, peaking in 1967 with approximately 170,000 PLA soldiers stationed in the country. Most of these troops served as technical advisors but many manned anti-aircraft installations.

The USSR sent over 2,000 military advisers and technical experts to North Vietnam over the course of the war, and some of these men also manned radar installations and anti-aircraft emplacements.

That represents the extent of contact between Chinese and Soviet troops with American forces in the Vietnam War. Any anecdotal reports of US special forces glimpsing Russian advisors on the ground leading North Vietnamese patrols or anything like direct ground contact is, at least to my knowledge, definitely unconfirmed and likely spurious.

For more background on Chinese and Soviet involvement during the war, Ilya Gaiduk's The Soviet Union and the Vietnam War, and Qiang Zhai's China and the Vietnam wars, 1950–1975 are both great sources.