Socialism and Communism are different types of economic systems that consider the fact that all resources in a place should be distributed equally among members and also create a central leadership system that will be controlling and monitoring the economy. However, the differences between socialism and Communism depend on the various practices of the two economic systems. Socialism system preaches equality among members of a state, and as a result, there is nothing like top class or low-class people in the country.
The leadership system instituted to be controlling the economy gives ultimate power to the people in deciding the direction they want the economy to go. Communism, on the other hand, also preached equality among members of a state and perceived all resources to be equally owned by all citizens of a particular country. However, the central leadership system instituted to be controlling the economy does not give the citizens the chance to decide on the economy, and hence, they act on the economy authoritatively.
I do not think there is any ideal Communist system in existence in the world. Most countries practice these two major economic systems: capitalism (an economic system which include the private ownership of resources or capital, with economic decisions made largely through the operation of a market) and socialism (economic system in which the means of production, distribution, and exchange are owned by the society).
Most countries use some combination of the two systems which is known as a mixed economy.
If by private property you mean personal property, communism allows for that. If you mean private ownership of the means of production, that wouldn't be allowed in communism.