What has a discussion about Joe Smith owning private property got to do with other people? So, if Bill Jones owns private property, it affects Bill Smith in what Way? How does Jones ‘exploit’ Smith.?
"Private property" is a social reality created by the rules of society, not a particular thing. And "private property" must exploit others or it is useless:
due to the dispossession of the vast majority of the population from the means of life, capitalists are in an ideal position to charge a "use-fee" for the capital they own, but neither produced nor use. Having little option, workers agree to contracts within which they forfeit their autonomy during work and the product of that work. This results in capitalists having access to a "commodity" (labour) that can potentially produce more value than it gets paid for in wages. During working hours, the owner can dictate (within certain limits determined by worker resistance and solidarity as well as objective conditions, such as the level of unemployment within an industry or country) the level, duration and intensity of work, and so the amount of output (which the owner has sole rights over even though they did not produce it). Thus the "fee" (or "surplus value"😉 is created by owners paying workers less than the full value added by their labour to the products or services they create for the firm. The capitalist's profit is thus the difference between this "surplus value," created by and appropriated from labour, minus the firm's overhead and cost of raw materials (See also section C.2, "Where do profits come from?"😉.
So property is exploitative because it allows a surplus to be monopolised by the owners. Property creates hierarchical relationships within the workplace (the "tools and equipment monopoly" might better be called the "power monopoly"😉 and as in any hierarchical system, those with the power use it to protect and further their own interests at the expense of others.
Within the workplace there is resistance by workers to this oppression and exploitation, which the "hierarchical. . . relations of the capitalist enterprise are designed to resolve this conflict in favour of the representatives of capital..." [William Lazonick, Op. Cit., p. 184]
Needless to say, the state is always on hand to protect the rights of property and management against the actions of the dispossessed. When it boils down to it, it is the existence of the state as protector of the "power monopoly" that allows it to exist at all.
So, capitalists are able to appropriate this surplus value from workers solely because they own the means of production, not because they earn it by doing productive work themselves. Of course some capitalists may also contribute to production, in which case they are in fairness entitled to the amount of value added to the firm's output by their own labour; but owners typically pay themselves much more than this, and are able to do so because the state guarantees them that right as property owners (which is unsurprising, as they alone have knowledge of the firms inputs and outputs and, like all people in unaccountable positions, abuse that power -- which is partly why anarchists support direct democracy as the essential counterpart of free agreement, for no one in power can be trusted not to prefer their own interests over those subject to their decisions). And of course many capitalists hire managers to run their businesses for them, thus collecting income for doing nothing except owning.
Capitalists' profits, then, are a form of state-supported exploitation. This is equally true of the interest collected by bankers and rents collected by landlords. Without some form of state, these forms of exploitation would be impossible, as the monopolies on which they depend could not be maintained.
For instance, in the absence of state troops and police, workers would simply take over and operate factories for themselves, thus preventing capitalists from appropriating an unjust share of the surplus they create.
Obviously, personal property such as a toaster isn't used to exploit others, so there is nothing objectionable about it.