Wouldn't communism be "our"?

no, it should be the State's pickaxe, that the central committee might entrust to you so that you work your part in the 5-year plan

It would be the state's pickaxe, but we'd call it "our" as that's the official nomenclature.

groan There's no helping you people. Communism is a society without a state, classes, or money. There's no such thing as a "Communist government".

Serious question: How is that different from anarchy?

Communism doesn't imply anarchy as it is primarily an economic system rather than a form of government.

how is a communist society organized, absent of any state? Does it self-correct, like free market is supposed to self-correct, according to capitalism?

I don't know why you're being downvoted, that's exactly how it'd work under Stalinist communist regime.

Stalinism is one specific variety of communism, in my experience most leftists today are critical of the overly centralized and rigid type of state structures that may have been present in historically socialist oriented nations.

This comment chain reminds me of the early 2010s

https://youtu.be/hWTFG3J1CP8?si=01CPwHIFDMoENWFe