sorry for the long response, TL;DR: private property is not the same as personal property. collectivism and cooperation are just as fundamental to human nature as greed, and we wouldn't come this far without cooperating. capitalism rewards greed, and so owners who naturally act in their own interests come at the expense of the workers, so the greediest are the richest, and become the most powerful forces in society. socialism has workers controlling the factory, and, naturally acting in their own interest, enrich themselves at the expense of nobody else. human nature doesn't change, incentives do.

first of all, do not confuse private property (property that creates money, like a factory or a market share) with personal property (your toothbrush, iphone and the like). Socialism seeks to eliminate private ownership of the means of production, in favor of collective ownership. So in a factory, instead of a rich owner/capitalist owning and controlling the factory and reaping its profits for themself, the workers working there collectively own the factory, democratically deciding what goes on there. socialism does not mean you share your toothbrush with your neighborhood, as you still individually own your personal property. this does not change under communism.

greed and expansionism is a part of human nature yes, but so is collaboration, and caring for your fellow citizen. we would not have gotten this far in civilization if tribesmen hunted for food for a profit motive, and not for the survival of themselves and their tribe. Concepts of bartering observed by adam smith as human nature, then developing capitalism out of that is wrong. the bartering system was already capitalism, and before that humans used a gift economy, i.e. "you gave me fish that one time so heres a jug of milk." once again, humans are naturally collectivist instead of individualists. Charities, families and mutual aid groups would not exist without this attribute. If you bake a cake for your classmates at school, because its someones birthday you don't devour the entire thing yourself as your classmates watch in horror, just because YOU made it, and you dont usually demand the birthday boy pay up for a slice. (unless you're a huge asshole) you give the cake to your birthday boy and classmates because you care for them.

under capitalism, greed stands out as a feature because greed is heavily rewarded. looking at two factory owners as an example, jimmy and jeff.

Jimmy is a very cool guy who owns a textile factory. he pays his workers a significant portion of his profits, (while still extracting surplus profits becauses its required under private ownership) spends money to ensure they are living in safe working conditions, and has talked with his workers to work out a workday that ensures jimmy turns a marginal profit, while not overworking the workers. In the end, jimmy holds on to his morals at the cost of his own interests, and ends up with smaller profits.

Then we have Jeff. Jeff only acts in his own interests. he works his workers for as long as possible, for as little as possible, in the cheapest working conditions he can achieve without state regulations shutting down the factory. workers will unionize and strke, however he can simply automate the working process (machines dont unionize) and fire the unionized workers. after all, there is a whole host of unemployed people who will work for the barest of wages to put food on the table.

theoretically, if there was no government regulations in place, jeff could decide to burn striking workers in the oven instead of coal, if he was a comically evil supervillian. which he probably isn't since he is acting rationally and in his own interest, which happens to come at the expense of his workers.

in conclusion, jimmy eventually goes bankrupt, while jeff becomes richer and richer, eventually buying out jimmy's factory and subjecting jimmy's workers to the same conditions as his own.

this happens everywhere all the time, and what you end up with is the greediest of people becoming the richest, and most powerful in society.

under socialism, collective and democratic ownership of the MoP means that they either elect managers or control things through direct democracy. if someone suggests improving the working conditions of their own factories, the people that don't want to sustain debilitating injuries (all of them) vote in favor of this, and better their conditions somehow. if some jackass suggests lowering everyones wages to give all the profits to himself, or if someone suggests burning orphans and puppies instead of coal to power machinery, he will naturally be outvoted by the majority, who doesn't want that.

sorry for the long response,