I'm currently reading Anne Applebaum's book, Gulag, and she goes over many similar scenarios in it. In short you would be arrested, sent to a holding cell and interrogated, sent to trial, then sentenced to either a remote camp or prison. From there you would either die a slow death of disease and malnutrition or survive.

Up until 1939 Russian secret police had immense pressure from the government, sometimes even Stalin, to not make mistakes. This often led to innocent people being arrested under mere speculation. There are even accounts of the NKVD admitting they arrested innocent people but were not allowed to free them. If you were actually found with contraband it was a guarantee that you would be arrested. In your scenario it sounds as though the NKVD were already in your home prior to finding the literature. Since they were already present it would be assumed they were there to arrest you anyway. Following your arrest you would be transported to a temporary holding cell and interrogated. Since they had hard evidence against you the interrogation process may be skipped. Next would come your trial, this could be up to years after your arrest. Despite all the chaos during the 1930s the legal system was still a vital part of society. Unlike Nazi concentration camps where people were not tried before exhile Soviet citizens received trials prior to being sent to concentration camps. These court cases were often less than 5 minutes in length however. In addition the person sentencing you was often a panel of 3 government representatives rather than a judge. These representatives were being pressured to sentence massive amounts prisoners as well. The reason for needing prisoners was due to the massive use of prison labor, yet prisoners were dying too quick for any efficient use of this labor force. So due to the fact you were already assumed guilty paired with the need for labor you would be sent to a remote concentration camp. The journey to the camp would most likely be in an unheated train cart and last for months. You would only be allowed to use the restroom once a day if you were lucky and begged the guards. You would not receive water for up to a week during this trip and food would be about 2 kilos of bread for the entire voyage. There would be no warm clothes supplied either and at the end of the voyage many people would already be dead. Also if you are a woman there is a good chance you would be gang raped by prisoners as the guards observed not intervening. Inside the camp itself you would be sorted as either fit for work, fit for light work, or not fit for work. If you were not fit to work you were almost guaranteed dead. The camps ran on a system where the harder you worked the more you ate. If you could not work you were only given 200 grams of food a day. The strong workers were given upwards of 700 grams of bread and around 100 grams of meat. In the 1930s there were many reforms taking place as well. If you were present in the early 1930s the guards could still kill you for no reason and report it as an accident. Towards the end of the decade Stalin implimented many new rules making prison life easier. Examples of these reforms are the introduction of no more executions (unless sentenced to death), better toilets, and mandatory exercise. Despite the attempts to make life in the camp easier you would still undoubtedly get lice and there was a good possibility of dysentery or typhus. If you somehow survived the camps there was also not a guarantee you would be set free once your sentence was completed. Often times poor record keeping meant prisoners were not released when they should have been.

I hope that answered your question. I'm on the road right now and don't have the book on me. When I get home I can find it and better cite some of these scenarios too.

Some questions to clarify OP's premise: Did the NKVD do regular searches of civilian homes or would there have had to be suspicion of a crime? Would that suspicion need to be based on anything substantial? And what qualifies as "forbidden Western literature"?

Directly to your question: arrest, unknown period of interrogation (from one day to weeks), 10 years hard labor. When the 10 years are up, if you are still alive - could be renewed by a few years by a special decree. There is a good chance that you would be in camps until 1953 - after Stalin's death.

Indirect answer - they don't have to find anything. you could be arrested because of something you did or didn't do before or during the revolution, because someone named you as an associate, someone said you did or say something.

If you read Solzhenitsyn - some of the reasons were just bizarre or ridiculous, like you smiled suspiciously or didn't clap hard enough in a meeting, or you used a newspaper with Stalin's photo as a toilet paper (there was no toilet paper in Soviet Union back then, and hard to get well into the eighties).

Sometimes, a local NKVD will get a request for a number of political arrests, and local interrogator will just tell you: "listen, this is what the party needs, so you should play along and confess to whatever BS we write up for you".