Communism aims to abolish private property. Market anarchism does support private property (I'm personally more in the Rothbard's range)

"Positions among free market anarchists can range: Mutualists believe in private property so long as it is based on personal occupancy and use; Cousins to the mutualists, the Georgists, envision a system of land-tenure in which people live in contractual communities whose public goods are financed from land rent; Rothbardians generally see rights to property as “human rights” that belong to all human beings."