In a Pure Communist Society, How Will Consumer Products Be Distributed Without Private Businesses?

Details like that differ depending on what school you ask. Most will say that the workers themselves will decide distribution and deliver the goods to the consumer. There are variants though that believe in a free market, look up mutualism.

Medicare communism

The government decides if you can have something.

The workers/peasants only get the barest necessities.

While the ruling elites can have all the luxury goods they want.

draw straws