I mostly come across it in literature of the period that deals with Britain, where food rationing continued for years after WWII was over-- things like bacon, butter, and sugar were rationed from 1940-1954. (Flour stopped being rationed in 1948; clothes were rationed until 1949; sweets, gasoline, and soap were rationed until 1950.)

So, for example, a farmer couldn't slaughter his own hog without a permit. (And ownership of a single pig may have been illegal?) So villages would have a "black market", of sorts, where a group of villagers might be "in the know". One of them might have a hog that he secretly allows to forage in the woods. The other villagers arrange for scraps to be sent to supplement its diet. Two or three other villagers might help with its slaughter/hanging/processing by loaning their hoist-and-gambrel in a quiet location, and setting to work on de-hairing and butchering it. And then the hog gets parceled out amongst the people in the secret group. But it was all very much against the law, and everyone cooperated in secret so as not to run afoul of the authorities.

But a little bit of digging indicates that Poland's big time of food rationing actually happened more in the 1970's-1980's, rather than in WWII proper (1939-1945).

"The first rationing, in 1947 and 1948, involved only meat and cooking oil. From 1951-1953, meat, soap, and washing-machine detergent were rationed... The real disaster happened in 1976. It started with sugar: two kilograms per person per month. In 1981, rationing expanded to meat and meat products. Then it was butter, flour, rice, and cereal. Next, it was alcohol, cigarettes, coffee, chocolate, and gasoline. Altogether, rationing lasted 8.5 years. Meat rationing didn't end until August of 1989."
I have a teacher from Venezuela who talks about how her family deals with the shortages there for years. For example, if they're planning a wedding, they start hoarding supplies a year in advance, so that they're able to give a good, festive party with yummy things to eat. And they spread it around a large group of family members, so that everyone contributes a little something to the celebration-- setting aside flour, or sugar, or powdered milk, or things like that. So, while it may be difficult to get enough eggs, for example, if you have 30 people all looking for extra eggs to contribute to the cause-- you're able to get enough for your needs.

I would expect the Poles to have a lot in common with the British and the Venezuelans, but there's a big difference between urban black market and rural black market. For example, with the Venezuelans, the farms were confiscated by the government, because people were only allowed to keep one home. And the government makes rotten farmers. So that made the traditional workarounds very tough in Venezuela-- because all of a sudden, private control of almost ten million acres' worth of farmland is out of the equation. But in a situation where farmers keep their land, they're able to do things in secret with a little cooperation from those around them.