Indeed, this did take place, and not just to the Soviet Union - Westerners defected to many of the other countries in the Eastern Bloc (mostly East Germany), as well as the People's Republic of China, and, at times, North Korea, Cuba, and Vietnam. Some defected to non-aligned and ostensibly neutral states, such as Switzerland and Sweden.

Wikipedia does have a relatively comprehensive list of (most of the) defectors from Western states here, but I can provide some additional context to the list.

Western defection and migration, especially from the United States to Soviet Union, took place before the Cold War, and, historically, was largely ideologically motivated. After the establishment of the Soviet state and the end of the Russian Civil War a number of Western Communist or Socialist sympathizers saw the Soviet regime as the manifestation of the revolutionary socialist government of the workers, the kind that many yearned for at home. Some of the Western socialists traveled to the Soviet Union both to document the transformation, and, in some cases, to hopefully facilitate it.

Perhaps the most famous example of such pre-WWII Western socialists is John Reed, the author of the famous book documenting the course of the establishment of the Soviet state "Ten Days that Shook the World". He died in Moscow in 1920, and is one of only three Americans buried in the Kremlin Wall Necropolis.

At roughly the same time the Soviet government embarked on a program of rapid industrialization, and as part of that program invited large numbers of Western engineers, some of whom elected to stay in the Soviet Union. Later on, during the Stalin purges many of those people would end up being tried for sabotage and conspiracy to overthrow the government and were either shot or imprisoned. Among those Westerners that came to the Soviet Union at the time were also some African-Americans, some of whom chose to stay in Soviet Union due to the racial discrimination they experienced at home (some of their children lived their entire lives in Soviet Union as a result).

During and after the Second World War this trend continued, with the people who either migrated (before the Cold War) or defected to the Soviet Union being either life-long Socialists or Communist sympathizers, or becoming enamored with the Soviet system during their travels or work. Some of the high-profile examples of Western defectors to the Soviet Union of that period include people like Bruno Pontecorvo and George Koval, both of whom were nuclear physicists and were involved in the Manhattan Project. Koval in particular turned Soviet sleeper agent, having officially received his Soviet citizenship prior to the war, and is credited for giving the Soviets crucial information that facilitated the development of Soviet nuclear weapons program.

The other high-profile case is that of the Cambridge Five, a ring of 4+1 (we are pretty sure about the identities of the first 4 agents, but not the fifth) British-born Soviet spies operating within various British security services during and after WWII. All of the known members were recruited as young men, playing off of either communist sympathies, or their opposition to the Third Reich (which was seen by many as the natural enemy of the Communist system), and later defected to the Soviet Union proper. Kim Philby, thought to be the initial member of the Cambridge Five, died in Moscow in 1988 at the age of 76, and has voiced his disappointment with the quality of life in the Soviet Union (especially with the way the state treated its WWII veterans), but allegedly never considered going back to the West.

Another extremely high-profile (in retrospect at least) case of defection is that of Lee-Harvey Oswald, who professed sympathies to the Soviet regime, and claimed to have been disillusioned with the life in the United States. That said, the Soviets were reluctant to accept him at first, and only relented after he attempted to kill himself, having been refused asylum initially. Oswald, like many other defectors, was able to enjoy a lifestyle that was considered extremely luxurious by Soviet standards, and yet paled in comparison to what you'd expect in the United States, so he, like many others, regretted his choice, and returned to the United States (a non-insignificant number of defectors couldn't bring themselves to go back and committed suicide instead).

As has been said, a lot of the later defections followed the same basic trends - people who were initially sympathetic to the Soviet regime or were Socialist activists (or members of the Communist Party of the United States), with a background in applied sciences. Most seemed to have had little actual knowledge of the situation in the Soviet Union at the time - or, rather, they held romantic views of the Soviet system, and either didn't know about, or outright dismissed the reports of state oppression, corruption, and low quality of life in the Soviet Union, even though some of them worked in intelligence agencies like the NSA. Accordingly, many ended up regretting their decision to defect later on. Anecdotally, a lot of the Soviet sympathizers, as well as some of the actual defectors, were referred to as the "useful idiots" in Soviet Union. The ones that seemed to have led relatively successful lives in the Soviet Union (and they seemed to have been in the minority) were mostly high-profile Soviet spies that were recruited by the Soviet intelligence agencies and were stationed in the West prior to their defections - career spies if you will. The Cohens, a husband-wife pair of American-born Soviet spies, were convicted for espionage in the States, but later exchanged for an MI-5 agent. Both remained active within the Soviet intelligence apparatus until their deaths, and their allegiance was never in doubt, which allowed them to find high-profile employment in the East (which was not the case with the majority of defectors to the Soviet Union).

By the end of the Cold War the defections from the West almost dried up, as the political climate in the United States became more tolerant, and the economic situation in the Soviet Union continued to deteriorate.