A number of prominent scientists working on stem-cell research have written a letter calling for peer-reviews to be made publicly available:
We suggest a simple step that would greatly improve transparency, fairness and accountability; when a paper is published, the reviews, response to reviews and associated editorial correspondence could be provided as Supplementary Information, while preserving anonymity of the referees.
In cutting edge fields such as stem-cell research it's very common for scientists to be working on exactly the same problem with the first to publish receiving the bulk of the plaudits. As a result, peer-sabotage can be a problem. Sabotage is not such a problem in economics because researchers rarely work on exactly the same problem but laziness and low-quality reviews are real concerns. (Note that my experiences with journal editors and reviewers has been more good than bad so I don't speak with sour grapes).
But would publishing anonymous referee reports really increase quality? Blog comments are public and anonymous and they can be stupid, rude, and ill-informed (not this blog of course). Indeed, the trend in blogs has been to remove anonymity as a way of increasing accountability and quality. In science, however, anonymity is essential because the opportunities for repeat play and thus collusion are too common.
The primary effect of published referee reports would be on editors who would have their work put under greater scrutiny. That is not necessarily a bad thing although editors are professionally under-rewarded in my view so more work is not necessarily going to lead to a better selection.
Note also that regardless of any change in quality, the value of the referee comments themselves should not be overlooked. Referees may also appreciate the opportunity to have their work published in some form. Early results from the EMBO Journal which recently switched to publishing referee reports online appear to be satisfactory. Thus, I favor experiments along these lines.
Here are previous MR articles on the peer-review process.
Addendum: Seth Roberts comments. See also Barkley Rosser in the comments.