What Plymouth suffered under communalism was what economists today call the tragedy of the commons [http://tinyurl.com/4xobe]. But the problem has been known since ancient Greece. As Aristotle noted, "That which is common to the greatest number has the least care bestowed upon it."
When action is divorced from consequences, no one is happy with the ultimate outcome. If individuals can take from a common pot regardless of how much they put in it, each person has an incentive to be a free rider, to do as little as possible and take as much as possible because what one fails to take will be taken by someone else. Soon, the pot is empty and will not be refilled -- a bad situation even for the earlier takers.
What private property does -- as the Pilgrims discovered -- is connect effort to reward, creating an incentive for people to produce far more. Then, if there's a free market, people will trade their surpluses to others for the things they lack. Mutual exchange for mutual benefit makes the community richer.