Alexei Yurchak's Everything Was Forever, Until It Was No More: The Last Soviet Generation discusses the cultural experiences of baby boomer/Generation X people in the Soviet sphere of influence and the role of bootleg recordings of Western pop music in shaping how such young people of the era saw the world.

As Yurchak tells it, argues that Soviet mass production of reel-to-reel tape recorders in the 1960s caused a sea-change in the ability of Russian people to hear Western music - "the circulation of Western music in the country increased exponentially". While good music, according to Yurchak, was meant to follow the 'healthy culture of common people', as opposed to the music that provoked 'unhealthy bourgeois instincts', youth in Soviet countries nonetheless gravitated towards Western music thanks to the proliferation of tape recorders allowing people to circulate music themselves, away from official channels.

The production of such tape recorders increased exponentially across the 1960s to the 1980s, and I get the impression from Yurchak's book that a critical mass in terms of Soviet youth awareness of Western rock music through mass taping was reached in the late 1960s or early 1970s thanks to such recorders. At this point the Beatles were soon to break up, and other bands that had other concerns to the Beatles, musically and culturally, were rapidly becoming more prominent, especially through the 1970s after the Beatles broke up and started releasing solo records to varying levels of success. So especially as the 1970s went on, The Beatles became 'starina' ('ancient stuff'). Yurchak records, for example, the experiences of Viktor:

Foreign records that were copied were brought into the country through port cities by sailors and diplomats, sold through networks of music lovers, and then tape-recorded thousandfold throughout the country. Viktor (born in 1959) became exposed to Western rock in 1972, as a thirteen-year-old school boy in the town of Smolensk. Every Sunday, recalls Viktor: “I walked with my huge ‘Kometa’ reel-to-reel recorder across town to the apartment of an older guy who always had new records. I recorded Black Sabbath, Alice Cooper, Bryan Ferry, paying him two and a half rubles per record. Most other people at that time liked the Beatles, but I did not care for them—who were they compared to Ozzy [Osbourne]?!

Overall, though, the Beatles specifically do seem to have played an important role for those of the generation Yurchak is profiling.

In making Western music their own, the last Soviet generation engaged in a complex process of cultural translation. When artist Dmitry Shagin painted a picture that was meant to represent the collective identity of that generation, he drew just two words on a canvas—TXE BEATлE3—a jocular, nonstandard, Cyrillic transliteration of the English words “the Beatles” that was meant to convey a Russian accent.

Overall, Yurchak sees the function of Western rock music as being part of 'the Imaginary West', a sort of imagined version of the West which was used in Soviet discourse which was something of a fantasy utopia:

When the external Marxist-Leninist canon for evaluating good and bad cultural forms became no longer determinate and authoritative discourse experienced performative shift, the constative meaning of these cultural forms as it was described in authoritative discourse became unanchored from form and could then change. Ultimately, this meant that one did not have to think of “socialist” and “bourgeois” cultural forms as inherently incompatible because their meanings could shift depending on how and where these forms were used.

This ambiguity of meaning allowed young Soviet citizens to inhabit a space where they could play with Western imagery and explore what they wanted in life, even where it diverged from the official desires they were meant to have, without too much official sanction. Later on, Yurchak discusses how the Imaginary West came to be part of everyday life:

[in] the 1970s and 1980s, [Western] nicknames were commonplace in schools and colleges, where students called each other Mike (for Mikhail), Alex (for Alexei), Bob (for Boris), Madeleine (for Elena), Margo (for Margarita)—nicknames that were not standard in Russian and recognizably Western....[despite their bland official names] many cafés became known as Saigon, Ol’ster (Ulster), Liverpul’ (Liverpool), London, Rim (Rome), Vena (Vienna), Tel’-Aviv, and so forth.

So The Beatles and their like became symbolic for this generation of a new way of relating to Soviet authority, one which saw a lot of absurdity in the middle ground between the repressive, authoritative state and the on-the-ground reality of there being considerably more room to move culturally than was advertised. So the Soviet citizens who came to see the world this way weren't doing so because of the Beatles and such bands, but because the Western rock fit into a useful space within the Russian cultural space for a generation who had a particular way of relating to government authority.