We tend to think of "communism" as starting with Karl Marx and the Communist Manifesto. Marx was certainly a significant political theorist, but he was actually part of an existing, much larger movement that has its origins in the same reforming impulse that lead to American democracy and many religious communes.
Today we think of "communism" and "democracy" as opposites, but in the 1600s, 1700s, and 1800s, they were more or less the same. Democracy meant a community of people had the right to determine their own laws, instead of the King dictating laws to them. Communism of the day was meant that the community had a right to determine their economic fates, rather than the King. In religious communes, in particular, councils voted on both legal and economic matters. Later, the idea of collectively running one's own country and of collectively running one's own factory can be seen as the same idea expressed at different levels.
As I understand it, our notion that the government and the economy are two separate entities is fairly modern. 500 years or so ago, the political and economic systems were completely entwined, via that system popularly known as feudalism.
I'm still not clear on when or why democracy/communism first became a big deal, but we see at least some interesting groups emerging in the 1600s, with a variety of systems. The Pilgrims of the Plymouth Bay Colony, in 1620, established a democratic society, apparently in keeping with Calvinist doctrines. The colony's government also administered certain economic concerns, like regulating the purchasing of land, but does not appear to have banned private property.
Some Quaker and Shaker groups did hold all property in common. The Diggers, around 1650, were agrarian socialists who attempted to farm on common lands. I believe the Mormons also practiced some form of centralized economic direction in the settling of Utah. And, of course, many monasteries and convents have been essentially communistic for centuries.
Some groups were obviously more successful than others, but overall, religious communes seem to have done pretty well, and may have provided much of the inspiration for the secular communism movement.