Anarchists (including us) mostly talk about personal vs private property. For example in an anarchist society nobody is going to take your toothbrush or house, but you aren't allowed to own a house you don't live in (yet still charge for) nor a factory where other people work, those things would be communally owned and cared for, or given to someone in need (in the case of a house). So it's kind of a semi-ownership at least compared to how it is now, you get what you need, not more than that.