The anarchist community remains divided over the question of property. The more visible Anarcho-Capitalist wing is all about private property, unaware or unwilling what that term means in practice, while the Anarcho-Socialist (etc.) wing doesn't really unpack that term properly and thus gets tripped up when attempting to argue for their form of anarchism.
I will attempt to unpack that term.
"Property" is the word, in English, that we use to refer to items or locations of a discrete nature that can be accounted as belonging to one specific individual or group. The laptop I'm using to compose and publishing this post is property. The blog that I published this post at is property. The body I used to transmit my thoughts from my mind to this post is also property. The house I reside in, the food that I consume, the books that line my shelves, and so on- all of this is property.
"Private" is the word, in English, that we use to refer to items or locations that are specific to an individual or group and not available without pause to the population at-large. The above property? Private, all of it. The roads I travel on, however, are public property. The airwaves over which I'm using WiFi is also public property. So are the parks I live nearby, and visit as I wish, and so are the libraries.
The issue is not "public" vs. "private". The problem is "personal" vs. "private."
"Personal", in English and referring to things, means that the item or location is particular to a specific individual or entity. In practice, we use this word to refer to items that one holds, wears, uses, or occupies. Anything more abstract in terms of ownership trends over to "private", in practice, as we see when talking of patents, copyrights, trademarks, offices, and so on.
What all anarchists agree upon is that personal property is a self-evident thing. One can, and usually does, own those items that one holds, uses, wears, or occupies. The clothes that I wear are my personal property. The laptop I'm using is my personal property. The knives I carry and use are also my personal property, so is the bed that I sleep upon. Therefore, what these two camps fight over is not about personal property. They fight over what is beyond the personal; they fight over what is "private" property.
In short, they're fighting over what is otherwise deemed "capital". They're fighting over mills, kitchens, workshops, hospitals, laboratories, and other places where work is done--something is made or a service provided--as well as the tools (including abstract ones, like theorems and algorithms) used to make that work possible. It's a fight over what is actually personal and what is actually public, for all intents and purposes; it's a fight over those goods and services, and the systems used to distribute and administer them- specifically, over an individual vs. collective paradigm of anarchy and how to make it work.
The problem, as I see it, is that both sides are ignoring their respective fatal flaws. The An-Capitalists want a system that, whenever it arises (and it has, albeit unknowingly so most of the time), inevitably produces a new feudalism; this is the experience of American colonialism in what is now the Southern Atlantic states (including Florida, eventually) as well as Carribean colonies. Because power and influence, in practice, stems from wealth and that is derived from ownership of private property it means networks of socio-economic tension between haves (landowners, ultimately) and havenots (everyone else). This leads to the creation of a State by the haves to create a barrier between them and the havenots. The An-Collectivists want a system that, whenever it arises, turns into a snitch-cum-police state as a body inevitably arises to "adminster" things and that becomes a power base in itself; the "anarchist" society becomes Statist when this body reaches critical mass and uses propaganda and leg-breaking to enforce conformity.
In short, the former produces Facism or Inverted Totalitarianism. The latter creates classic Totalitarianism. (Or, if you will, Abusive Father-as-State vs. Abusive Mother-as-State; both are toxic and democidal.) Neither of these creates a sustainable anarchy. The fight over property, quite frankly, that they do cannot end any other way. That doesn't mean that there isn't another way out.
Acknowledge the difference between "personal" and "private". Personal property is self-evident manifestations of Natural Law. Private property is the fiction shat out of Man's ass, and inevitably requires the creation and perpetuation of the State to keep the illusion up and running; it has sweet fuck all to do with Natural Law. A strict individualist stance on what property is, and therefore how to deal with it, is how you navigate these seas between Scylla and Charybdis. Disavowal of both Capitalist and Collectivist ideas will go a long way towards putting down the problems that exists in Anarchism right now; instead, stick to the personal- "If you don't hold it, then you don't own it." is a very useful guideline to follow.
Working towards an alternative paradigm for when Empire falls.