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Say a reviewer writes something like

This paper is [yada yada blah blah].

Sincerely,

Professor John Smith, Big Name University

Should the editor just forward the review to the authors because Professor John Smith, by signing his name onto the review, is presumably willing to reveal his identity to the authors? Or should the editor keep the review anonymous by deleting the signature?

If the journal is structured with a blinded review process, as most are in my experience, I would censor the name as an editor.

Only if there is some sort of explicit journal policy allowing reviewers to unblind themselves would I consider not censoring the name.

There are a few different cases to consider. First of all, there is the question of whether a journal's policy even allows for signed reviews. I think that most journals do not have an official policy about this. However, if there is a strict prohibition against non-anonymous reviews, then the editor should remove the identifying information before sending the report on to the authors (and any other relevant parties, such as other referees who are working on the same paper).

In the more likely event that signed reviews are not outright forbidden, then editor should look at the additional question of whether the referee really intended to make their identity known. From the report alone, it may or may not be clear whether a referee is intentionally choosing to dispense with anonymity. If there is just a signature at the end of the report, the reviewer might have added it out of absentmindedness. If the situation is unclear, the editor should check back with the referee, to see whether they actually intended to include their name before passing that name on.

However, I have seen one review that concluded with:

I choose to sign this review.

[Referee's Name]

In that case, it was quite clear that the reviewer (who was both a very senior person and giving a positive report) was not worried about maintaining anonymity. In a clear-cut situation like this, a referee can simply send the authors the report without any additional concerns.

So, I have heard of people not censoring when people do so. There have been at least some controversies in some fields where this has happened. See for example, https://www.quantamagazine.org/a-self-aware-fish-raises-doubts-about-a-cognitive-test-20181212/ . I would however strongly recommend removing the signature. Anonymity is important, and I personally (and other people) have had bad experiences with referees who have deliberately unmasked themselves. Also, it is possible that the file you got was intended for the editor and wasn't actually intended to not be unmasked in the first place. But regardless, editors should do all they can to keep the referees anonymous.

If the journal policy is to maintain anonymity then it should not be done, even if the referee has indicated her/his name can be revealed. The reason is simple enough: if the review is not signed and one knows that John Smith from Big Name University usually signs his reports, then one can deduce the referee was NOT John Smith, which may help the author conclude about the identity of the real referee.

"Should" or "should not" is impossible to answer in the general case. Some journals may have a formal policy one way, some may have a policy the other, and I suspect that the vast majority have no formal policy about what to do with signed reviews.

There is a (small and localized but real) debate over whether reviewers should sign their reviews, and at least for a while it seemed that there was a small movement toward signed reviews. I can say that I've signed reviews and at least in some instances they have been passed on to the authors, so there are some journals and editors who don't have concerns about this.

I think you should delete it. Otherwise you remove the puzzle game of figuring out who reviewed you. (Easy one is when they ask you to cite them.) ;-)