That said, I think there are various answers.
One is to have a militia to enforce those kinds of principles. Anarchists who are also pacifists naturally reject this option.
Another option is to claim that in a society where everyone's needs were met, there'd be no desire by anyone towards private property, and that the desire for private property is a historically conditioned product of capitalism, and feudalism before that.
Another argument is that it wouldn't much matter as long as the default state was no private property. John T. Private can take his guns and go live in the woods if he wants to, what is essential is that the majority of property is public and therefore private property does not give a specific class control over the output of others' labor.
Another position, and this was Proudhon's, is that there is a distinction between personal property and private property. Proudhon would actually say that a family's household is an empowering, important kind of property which must be protected and preserved. Ownership of capital, however, Proudhon objected to, because it allowed, in his view, for those who owned the factories, machinery, mills, and plantations to present those who had no stake in the means of production to either work for one of them or to fend for themselves. As industrial society developed, fending for yourself because less and less plausible, and eventually the choice was between selling the output of your labor for (a socialist would say) less than its cost, or dying, basically.
Calling that choice barbaric is the essence of the social-anarchist critique of capitalism.