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Consider the scenario where an author sends his paper to Journal A. Journal A then sends the paper to reviewer X and the reviewer rejects the papers. After rejection, the author send his paper to Journal B. Journal B then sends this paper to reviewer X.
Does this ever happen? What does the reviewer do in this case?
It happens all the time. I think it is the reviewer's responsibility to respond to the editor saying that he or she has already seen the paper. Then it's the editor's call. Ideally the paper would be sent to a new reviewer, but depending on the subject of the paper the pool of available referees might be small enough that the editor requests the reviewer to reread the paper anyway.
When I, as a referee, see a paper that I've already reviewed, I just send the second journal the same report. Generally, minor changes are required, but I generally try to keep the report basically the same, so that the authors will know that the report came from the same referee. I can't say I have a particularly well-thought-out reason for doing this. However, I do think it serves to make clear that multiple editors have seen fit to pick somebody (that is, me) as an expert who is qualified to review the manuscript; the reports are not coming from people who were unqualified to referee the submission and should never have gotten it in the first place.
I do not inform the editor about this unless there is a specific reason. Since these situations pretty much always involve rejected manuscripts, the authors sometimes get angry in their response letters. If I feel that the authors have behaved inappropriately during the first round of submission, I might mention that to the second journal's editor. Similarly, if I rejected the paper the first time due to outright plagiarism, I will mention that to the second editor as well. Whenever I have done this, it has been to give the editor a specific heads-up and there may be abnormal difficulties with the paper.
I am also more likely to recommend outright rejection by the second journal if I recommended major revisions at the first. If the authors were unwilling to make the revisions to get the article published in the first journal, it's probably wasting people's time to suggest publication may be possible after the same revisions the second time around. Once, I had to recommend rejection of a paper at the first journal even though the authors had made useful changes. (They fixed one of the two serious problems but were unable to deal with the other.) When I got the paper to referee again at a second journal, they had reverted to their original version of the manuscript, with both problems present again. I was particularly unsympathetic in my report in that case, since the legitimate efforts I had made to help the authors improve their paper had evidently been tossed away. (I think I informed the editor in that case as well, but I'm not certain.)
This happens a fair amount in my field. Sometimes, it is even arranged. The author(s) tell Journal B that Journal A rejected the paper, and the editor at Journal B asks the editor at Journal A who Reviewer X was so that they can use the same Reviewer. Of course the authors are not stupid; this only happens when the paper was originally rejected solely for not being good enough for Journal A and Journal B is a less prestigious journal.
It is good to let the editor know (if he or she does not know already), but generally, having reviewed the paper before is a good thing. It means that you can submit a report quickly rather than taking 6 months or a year.
