= Private Property is both excludable and rival. Private property access, use, exclusion and management are controlled by the private owner." 
Steven N. S. Cheung:
"Any productive resource is a private property if, within well-defined limits, its owner has:
(i) the right to exclude others so that he alone may decide on its use;
(ii) the right to extract exclusive income from its use; and
(iii) the right to transfer or sell the property or resource to any one as he sees fit." (http://opensource.mit.edu/papers/chiao.pdf)
Private property can never be absolute
Donald P. Goodman III, expressing the social doctrine of the Catholic Church:
"The first is among the most important, the most basic, and the most hateful to a capitalistic mind: private property is not subject to the sole and absolute control of its owner. Indeed, while ownership of property is certainly a right in all civilized states, use of private property is subject to just state and community regulation.34 For this reason, Leo XIII taught that “the individual and the family should be permitted to retain their freedom of action, so far as this is possible without jeopardizing the common good and without injuring anyone”35; clearly, if it does jeopardize the common good, their freedom of action with their property should be properly limited. And further, the great pope taught that the state cannot forbid private ownership, but it can “control its exercise and bring it into conformity with the commonweal.” (http://distributistreview.com/mag/2010/07/individualism-and-the-state-part-ii/)
Kevin Carson's market anarchist view
"First of all, fixating on the specific words “private property” without regard to what is meant by them is a notorious example of mistaking the map for the terrain. As I’ve argued before, virtually every society in history, going back to pre-state agrarian communism, has had “private property” in the broad sense of a set of rules regulating priority of access to finite goods. If you don’t believe it, just imagine some random person in an anarcho-syndicalist society wandering into Machine Shop No. 17 and fiddling around with a lathe without permission.
On the other hand, most principled systems of property rules for land make a distinction between legitimate and illegitimate property rights. The rules for constructive abandonment or transfer may vary, but all of them treat the enclosure of unoccupied and undeveloped land as illegitimate. Thomas Hodgskin, a sort of English version of Benjamin Tucker, used the terms “natural” and “artificial” rights of property in a way that pretty much overlapped with Proudhon’s distinction between “possession” and “property.” Tucker himself sometimes used “property” as a pejorative, in Proudhon’s sense, and at other times spoke of an anarchist society in which legitimate “property” was based on occupancy and use. Franz Oppenheimer and Albert Nock made a similar distinction between (respectively), “natural appropriation” and “political appropriation” of land, and “labor-made” and “law-made” property.
My own preference is for the occupancy-and-use standard of Tucker and J.K. Ingalls. I have written extensively in defense of natural resource commons, and of the communal open-field model of land ownership that was almost universal from the Neolithic to early modern times. Other C4SS writers from a Rothbardian background take a more Lockean position, but even here most of them have moved considerably leftward from Rothbard’s own position and embraced much more liberal standards of constructive abandonment and adverse possession that, at the very least, considerably blur the boundary with occupancy-and-use. C4SS actually hosted a Mutual Exchange symposium on property rights that included advocacy for a wide variety of standards, with most of them falling closer to occupancy-and-use or Georgism than to the standard an-cap version of absentee ownership." (https://c4ss.org/content/46875)