There was always a strong interest in western entertainment products in the Soviet Union during the Cold War. Unfortunately for them, there wasn't a widely held form of distribution for visual media - VCRs only got rolling in the US in the early 80s, and probably didn't make very deep penetration in the Soviet Union before the fall came.

One form of easily available form of western entertainment was music, since records were easy to transport and play. Of course, in the 60s The Beatles were as popular in the Soviet Union as anywhere else on the planet, and demand for their recordings far outstripped the supply. People discovered that they could etch the music into discarded X-Rays, making some of the most unusual Beatles artifacts - "The Beatles on Bones."

Follow up question: How large was the black market on western produced products, if there even was one?

Do you want to extend the question to the Soviet Bloc?

Throughout most of Soviet history it was impossible to watch movies on your own, because technology of VCR/VHS wasn't there yet. It changed in the second half of eighties. I've flipped through some online archives of newspapers:

The Chicago Tribune, March 1986:

"The government newspaper, Izvestia, reported that a dozen black-market video dealers were put on trial in the Baltic city of Riga, charged with copying films for paid screenings in private flats and run-down buildings. Judges studied the 415-film bootleg library before sentencing the ringleader to 18 months in prison. His specific crime was charging $13 per person to view pornographic films."

Source: http://articles.chicagotribune.com/1986-03-17/news/8601200130_1_vcr-market-bootleg-tapes-video

The Inquirer, November 1986:

"Video Salon carried only one U.S. title: Convoy, a 1978 trucker adventure starring Kris Kristofferson and Ali McGraw. "It's very popular," said the clerk.(...) Of course, there is always Stallone. The Party has generated much propaganda denouncing Rambo and Rocky IV. Soviet poet Yevgeny Yevtushenko made a speech trashing Stallone's insulting two-dimensional portrayals of Soviets. But that only boosts sales for Stallone on the black market."

Source: http://articles.philly.com/1986-11-27/news/26092993_1_zhiguli-rubles-soviet-black-market

The Evening Sun, January 1991:

"However, the tastes of American and Soviet audiences are not always the same. For example, "Batman," which had great success in U.S.A., had bad luck in Soviet Union. (...) Soviet spectators know very well Jack Nicholson, Jessica Lange, Robert Redford, Al Pacino, Robert de Niro, Dustin Hoffman, Sean Connery, Mickey Rourke, Sigourney Weaver, Cher and many others." And he mentions that Schwarzenneger and Stallone are very popular in Russia.

Source: http://articles.baltimoresun.com/1991-01-04/features/1991004130_1_soviet-union-hippies-movies

Los Angles Times, June 1991:

"Of the 500 titles the salon maintains, the most popular are commercially oriented American films: "Tango & Cash," "Total Recall" and the "Police Academy" series."

Source: http://articles.latimes.com/1991-06-15/entertainment/ca-528_1_video-pirates

And here is story about bootleged movies in another Communist country, Romania: http://www.openculture.com/2014/02/how-bootlegged-american-movies-helped-defeat-communism.html

I'm kinda late to the party but can give you my two cents (and, maybe, give a little extra information about Video Salons mentioned in another comment).

It's from my personal experience mostly but I was somehow involved in this business.

So, before the VCRs it was almost impossible to watch US movies that were not released to soviet cinemas (and most of them weren't with some exceptions). But from the mid-80s it became much easier and cheaper because there were VCRs — too expensive for the most of us (at 1988-1989 you could easily exchange new japanese VCR for the russian car), but still accessible through Video Salons. Video Salon is a very small cinema with color TV and VCR attached to it. Almost all of them were completely illegal (maybe there were some exceptions that I don't know of but I don't think so) — all movies were pirated and so on but authorities were ok with that kind of business. So you are naturally buying a ticket, and you are in the small dark room with 10-30 other persons, and you watch some action movie, or chinese/taiwanese kung fu movie, or even erotics. All movies were voice-translated by unprofessional translators-narrators, and some of them became incredibly popular — even now, 20-30 years later, a lot of russian people knows the name of Leonid Volodarsky and his voice although almost no one could describe how he looks.

The prices were comparable to cinema prices, maybe slightly higher (usually it was 1 rouble per movie; cinema tickets were priced from 0.1 to 1.6 roubles depending on time and movie). Here you can see the handmade poster/schedule for the Video Salon (http://img11.nnm.me/e/d/3/e/d/f3699f8c7c71c3874893821c619.jpg). The name of this Video Salon is "Record". At 17.00 — Tom & Jerry cartoons. At 19.00 — movie with Jackie Chan. At 21.00 — Alien. At 23.00 — Lemon Popsicle (israeli erotic comedy; extremely popular in Soviet Union at that time). Price of entry — 1 rouble.

here is another one (http://img-fotki.yandex.ru/get/9767/12959330.a/0_e8c3c_86e86ecd_orig). I won't translate all the movie names but just one: "The 1000 eyes of the ninja".

And here you can see the pirated VHS tapes (every one consists of two movies or movie plus cartoons): http://byaki.net/uploads/posts/2015-02/1424118475_00.jpg

The picture quality often was quite bad, because someone can't copy VHS without degrading quality of the picture. So, there was some professional slang for this — original (master copy with translation; it was used mostly for doing copies), "the first copy" (quite good picture quality), "the second copy" (not so good but watchable), "the third copy" (quite bad but still watchable if you have a good imagination).

Video Salon golden age was short. VCRs became cheaper, and from mid-90s there was another trend — Video Rentals (with the same pirated movies and the same translators).