A review of the usual referee process: The editor-in-chief of a journal receives a submission and assigns a member of the editorial board to act as editor for that paper. The editor will contact potential referees and ask them to review the paper. The referees read the paper and write a report on whether to recommend the paper for publication and give suggested changes. They send this report to the editor who relays this information to the authors without revealing the identities of the referees. The authors will comment on the referee reports and revise the paper. The paper often goes back to the referees and the process repeats until the editor is happy with the paper or decides the paper is not worthy of the journal. The editor then sends his recommendation back to the editor-in-chief.
An argument for posting the reports goes as follows: If we make the reports public (though still anonymous) referees will feel a greater responsibility to make their report thorough and accurate.
The process of refereeing requires considerable back and forth conversation between the three parties: the authors, the editor and the referees. Posting the original reports will give a misleading view of the process and will cause referees to act far too cautiously about pointing out problems in a paper.
We have an editor responsible for making sure that he has enough quality reports to properly make a decision about publication. Using the public as an editor will often lead to some very difficult and messy situations.
Keep in mind the distinction between the referee process and reviews of published papers. We should encourage publicly available reviews of research papers whether in Mathematical Reviews or informal settings like weblogs. These reviews help us find the gems among the sea of research papers out there.