I guess it depends on the definition of private property and communal ownership. For me private property is synonymous with private ownership, meaning someone, not everyone, owns the means of production. With communal ownership I don't mean communal society, but rather ownership by a group. For example everyone within a company owning the means of production of said company would be communal ownership.

In that sense I disagree with coops not preventing individuals owning private property in the means of production, because that creates the same frictions that capitalism creates. If someone owns a part of the means of production that you need to create the product, then it creates a situation where they can use that ownership as leverage because without that part you cannot create the product. If their means of production are irreplaceable then they can demand more fruits of labor, because they own a key part of productions, which is essentially what capitalism is. If it's communal ownership there's no leverage because it's for everyone to use and nobody can demand more simply because they own a key part of the production.