The Moscow International film festival would screen films from the west that would not be available to the wider audience - open to the public, screening was for smaller audiences; the festival was held every two years from 1959 onward; Students could attend these screenings.

Also he was a member of the Soviet Film-makers union this union along with the writers union was in a very privileged position; so he could see some western films on its premises; however that would not have been possible for mere student.

After Tarkovsky, in the eighties some people who were permitted to travel brought in VCR recorders, along with the movies ; but that was later...

Edit: Of course regular cinema had some western movies also, (see comment of confused_druse); also interestingly Soviet movies ran lots of Indian movies.

It's hard to characterize the whole nature of Soviet film censorship; it varied greatly between Soviet regimes (late Glasnost was comparatively free, but definitely not the post45 Stalin era). But I get a sense from your question that you're examining the Thaw onwards. In essence, the USSR arranged for import-export deals for cinema outside the Soviet bloc. The main agency Soveksportsfil'm would arrange to buy packages of films in exchange for a similiar deal with Soviet films. Other times Soveksportsfil'm would buy films directly. Censorship was very heavy-handed, but less political than one might imagine. Violence and sexuality became more frequent targets of censorship than political themes, especially since the politics of Western cinema from the 60s onwards became harder to classify according to classical Marxist precepts. This was one of the reasons why Soveksportsfil'm preferred Bollywood productions because they didn't have the same level of difficulties for the censors.

It's important to realize that the Soviets emphasized film as a tool for cultural improvement and uplifting the general populace. This was why much of the direct contact Soviet filmmakers had was less with mainstream Hollywood but with the non-American art house set. Kurosawa's 1975 Dersu Urzala is an example of how Soviet cinema connected with various filmmakers. Mosfilm was able to give Kurosawa something he could not get in his native Japan: funding. After the box office failure of Dodes'ka-den, Kurosawa could not get financial backing for his films and Mosfilm stepped into the void.

One tangential way that the Soviet public became aware of Western film was through the magazine Soviet Screen. It would offer surveys of Western cinema with the intent of educating its audiences about it, but oftentimes it made Soviets aware of such icons like James Bond or Marilyn Monroe. Of course the articles would highlight the bourgeois cultural degeneracy and frivolity, but it exposed Soviet readers to these new cultural norms.

And given the privileged nature of cinema as a cultural institution, film schools had access to a wide range of Western cinema that mass Soviet audiences did not. Kieslowski at the Lodz Film School describes the experience as such:

Of course we weren't shown James Bond fighting the KGB, but we did see films which weren't generally shown in Poland or we saw them long before they were shown. I don't think there was any political censorship in their choice of films... The school just wasn't tinged with Communist propaganda. It was really open-minded, and that's why it was so good up until 1968.

Granted, Kieslowski's experience in Poland was different than in the USSR since Poland had a degree of cultural autonomy by the late 1960s, but his experience was indicative of a wider trend among Soviet and Eastern bloc filmmakers. The exposure to European and non-Hollywood cinema tended to push their aesthetics away from the haute cultural norms of the Soviet establishment.

Sources

Kieślowski, Krzysztof, and Danusia Stok. Kieślowski on Kieślowski. London: Faber and Faber, 1993.

Menashe, Louis. Moscow Believes in Tears: Russians and Their Movies. Washington, DC: New Academia Pub, 2010.

Roth-Ey, Kristin. Moscow Prime Time: How the Soviet Union Built the Media Empire That Lost the Cultural Cold War. Ithaca: Cornell University Press, 2011.

As other here noticed, Soviet people had seen Western movies. My parents mostly remeber comedies (often French) and Western movies were rare occurances everyone looked forward to, but still. Also with VHS taking hold in USSR people watched and copied many different movies. It was also heavy censored. I remember how in 90th Russian TV Channels showed special edition of some classic movies like Gone by the Wind, Some Like It Hot or Caesar and Cleopatra with mostly Soviet dub plus scenes that were censored. Censored scenes where mostly "amoral": bedroom scenes or candid jokes.

Also it seems that Soviet movie reviewers based their reviews on Western press. This review of Star Wars cites some French newspapers and it's rather obvious that author hadn't seen the movie.

Another thing is Tarkovsky himself said he was influenced by Kurosava and Bergman.

Yes, they did have opportunity to see many films that were not released, though they were mostly mainstream festival level movies. Many people in the film industry knew about Fellini, Antonioni, Godard, Kurosawa, Bergman, Coppola, Kubrick, Bunuel and other directors who could be ranked near them.

Somebdoy even bothere to compile a list of Tarkovsky's opinion on about 50 movies from his diaries, most of which were not Soviet and were not released in Soviet cinemas. (Not surprisingly he didn't like most of them)

It's not entirely accurate that Soviet Union did not import foreign movies. For example, the Magnificent Seven was one of the most popular movies among teenagers, as well as many Eastern bloc movies starring Gojko Mitic, whose genre can be described as "Serb playing Native American beats the shit out of cowboys". As far as I recall, Fantomas series was also very popular. I'm almost sure the Soviets had to release the Count of Monte Cristo with Jean Marais.

My father also recalled that he has seen The New Centurions (1972) around mid 70's, which impressed him very much at the time (he was around 15 at the time).

Might be a bit late on this one but they weren't just allowed to but they were often expected to keep up with international trends and important releases even if they weren't deemed fit for public consumption. The USSR took culture seriously and did it's best to ensure film that made waves internationally was made available to those within the filmmaking community. It was sort of a double standard, they wanted their filmmakers to be able to keep up with the techniques of foreign films while shielding the populace from things that might be considered subversive.