There are indications that at times early Christians shared common expenses and shared material possessions. (For just one primary source, check Lucian, Passing of Peregrinus, 13). To a modern person, this may indeed sound like communism.

Lucian often portrayed Christians as radicals in his writing, and this was probably a common belief. Yet I think the Roman authorities were much more concerned about the radical Christian theological beliefs than they were of any economic theories that we might consider similar to communism. As far as I know this "communism" didn't concern them in the least. Sources from authorities, such as the letters between Pliny and Trajan, seem to support this.

It is also important to point out that there are questions exactly how widely this "communism" was practiced. Although this deals with a later period than the 2nd Century Lucian, a fascinating book on Christianity and wealth is Peter Brown’s Through the Eye of a Needle: Wealth, the Fall of Rome, and the Making of Christianity in the West, 350 – 550 AD. By this time the Church was well established, and rich Christians were less likely to give their wealth to the "community" and more likely to give it to the Church to do as they saw fit. Yes, much of that money went for social welfare, but not all of it did. By this time there was a Church hierarchy which spoke for the community and less of a populist movement.