What has been the reception of Posner & Weyl’s claim that property rights lead to market power? Posner &amp; Weyl claim that property rights lead to monopoly rents:
Property Is Only Another Name for Monopoly
Radical Markets
Have their claims been elaborated, tested, or peer-reviewed?
In situations with assets with unique attributes that cannot be substituted (e.g. a plot on a high street) this seems to be intuitive. An owner cannot be compelled to subject her property to market competition but she can hold out for a higher price than she values the property at. This generates an economic rent resulting from her market power over the plot of land in question. The claim seems to be less obvious where assets can be easily replicated (e.g. a pint of milk). That would imply that property rights may only lead to monopoly rents where they grant property over assets that generate monopoly rents (e.g. land).
In narrow sense of the word the article was definitely peer reviewed as it was published  in Journal of Legal Analysis. But in this narrow sense it was most likely peer reviewed by jurists not economists.
In a broader sense of the word there are economists who had a look at the claims as well. For example, review of their book by Levine (2020) in Journal of Economic literature (one of the top ranking journals), examines their claims about monopolies and also other claims and policy proposals made in the book. The review is not very warm to the ideas of Posner and Weyl. Regarding the idea of property rights causing widespread monopoly power and misallocation Levine writes [emphasis mine]:

The chapter begins with a clear and careful description of the hold-up problem.
Here, indeed, monopoly power is a problem.
It then goes on to suggest that the hold-up
problem is ubiquitous in ownership of property—that all property is monopoly and that
the resulting misallocation might be on the
order of 25 percent of income. Here as elsewhere the book is not strong on data. There
is extensive research indicating that in some
places at some times, misallocation is a problem possibly of this order of magnitude.
Typically this research does not attribute
misallocation to a problem with private property—generally the reverse.


... Nevertheless the economics
profession does not generally view hold-up and private information problems as ubiquitous nor externalities as insoluble. In many
markets, ample substitutes are available—
for example in the housing market—so that
monopoly power is not important. In some
cases the market solves the hold-up problem relatively well—through tender offers
and other ways of packaging properties.


Pushing on: the authors argue that the
problem with the ubiquitous monopolies
created by private property can be solved by
a self-assessment wealth tax with mandatory
sale at the self-assessed price. Self-assessment
with a commitment to sell at that price is
a useful and credible solution to problems
involving assets for which there is substantial
private information. This has been a particular problem for commercial real estate and
for privately held businesses, where often
artificially low transfer prices are used to
avoid taxes. As the authors correctly say, top
researchers such as Harberger (1965) and
Cramton, Gibbons, and Klemperer (1987)
have argued for self-assessment in these
types of circumstances.
It is not immediately clear why a proposal
for solving a particular private information
problem should be viewed as a broad solution to a problem of ubiquitous monopoly...

There is more critique on this in the paper. To give you a wrap up (as mentioned in the comments under your other question), the problem with Posner and Weyl is not that their analysis would be inaccurate in some specific cases. Rather they try to generalize well understood result that holds in some very specific circumstances onto property rights in general.
There is quite large consensus in literature that property rights do not generally lead to monopoly or even market power (see Belleflamme &amp; Peitz Industrial organization: markets and strategies for overview of many models where even monopoly does not guarantee market power and models of competition where property rights are implicit but firms have no market power). In fact as Mankiw put it in his Principles of Economics, economists view property rights as &quot;an important prerequisite for the price system to work&quot;.
Furthermore, Acemoglu and Robinson (top developmental economists) in their magnum opus Why Nations Fail, go even so far as to state:

To be inclusive, economic institutions must feature secure private property...

Acemoglu and Robinson go further on proving that sustainable growth and economic development without inclusive institutions (including property rights) is virtually impossible.
Additionally, this work is apparently even criticized in law profession itself (see Wyman, K. M.; 2019), but as I am not a law professional I will have no further comments on that (it just seemed relevant to point that out).
Hence given the above I think it is fair to say that the claims of Posner and Weyl are not received positively by economics profession, to put it very generously.