Anarchists and communists make a distinction between personal and private property.
Private property refers to private ownership of the means of production, which is used to coerce laborers into working for a wage so that the capitalist can accumulate capital earned by the worker's labor.
Those who possess private property; capitalists, use violence to enforce it, either directly (e.g. private security, private union busters) or via the state's systems (e.g. the police, the courts, the military).