Proper way to deal with papers you've already refereed. This question is anonymous for obvious reasons.
I referee what feels like a decent number of papers (though I don't know how many is normal!), and I try to take it seriously.  Sometimes, based on something I explicitly said or something implicit in my review, the journal rejects the submitted paper.  Usually this seems to be because the editor feels that the paper isn't up to the journal's standards.
In an event that has proven less rare than I would like, I am sometimes asked - by a different editor and journal entirely - to review the same paper again.  My record is currently three such rotations.
Usually I try to beg off, because I feel like the author deserves to get a new referee who is less biased and might bring new insight to their review.  What I've found in practice is that editors tend to persist, and solicit my opinion more informally.  I understand this, because I know that referees are hard to find, and so I've usually done so.  The editor usually will say that they will seek out an additional referee, though I rarely hear the final story.  In some instances I have gotten the impression, for reasons that I don't want to get into, that an author is starting to feel persecuted due to repeated rejections by prestigious journals.
What is the best policy on requests to repeat a refereeing job?
 A: I agree with the side saying that you should do the refereeing for the second, third, etc time if you are asked. In particular I very much agree with no name and Ben in the reasons they cite. 
I would add the following. Very often there are only a few people who can truly evaluate a paper's worth. If you are one of those and you decline refereeing the chances of the paper going to someone who has much less perspective on the field is higher. Then some decision will be made, possibly taking much more time and it is not necessarily a better decision than what you would have suggested. If independent editors pick you as the best choice to do the evaluation then you probably are (at least one of) the best to do it. 
I suppose that when you decide to suggest rejection of a paper you do that based on the quality of the paper versus the quality of the journal. When you are asked to do the same for a different journal this comparison changes and so it is possible that you would feel that the paper is appropriate for the second journal and suggest that they accept it. You could actually try to influence this in your initial opinion. When you suggest that they reject the paper you could give an example of a journal that you feel is appropriate.
Regarding the statement that submitting to a second journal is like asking a second opinion I am not sure that I agree. On one hand we would all like to ask for a second opinion when a paper of ours is rejected, but fortunately we cannot do that. However, we would still like it to be published, so I think it is more like a second shot then a second opinion.
A: To add a little balance to what Felipe and Pete say (and comparable to what Ben says), if you feel your paper has been acceptably refereed modulo the issue of whether nor not it is appropriate for the journal that it was rejected by: meaning you're confident the referee understood the paper and the context, then when you submit to a 2nd journal it makes sense to mention that it was previously rejected, and give the name of the journal it was rejected by.  This makes the editor's life much easier and I think it's appreciated.  If you feel you're being persecuted I suppose this isn't a route you're inclined to take.  But then again, if this route succeeds you'll quickly discover you are not being persecuted. 
A: I am answering this anonymously, also for obvious reasons. As a referee, I get quite annoyed when, after having recommended rejection for a paper, and given several detailed reasons for the rejection, the same paper comes back to me from a different journal with no changes at all. In other words, the authors didn't even try to address my objections. In this case, I have absolutely no qualms about rejecting it all over again. On the other hand, if the authors genuinely tried to answer your objections, and you still think it is not acceptable in the new journal, maybe you should ask the editor to send it to a different referee.
A: My personal feeling is that not making the author wait for a second referee to write a report is doing them an enormous favor, even if the paper gets rejected.  If you aren't going to write the paper a good enough report to get the paper published, most probably whoever else would referee it instead won't either, and they will take a lot longer.
I have to admit, I also feel like you're not respecting the time of the anonymous person who will have to referee the paper instead of you.  Peer review takes up a lot of mathematicians time; it seems like just making the problem worse to create unnecessary duplication.
A: Presumably, whichever field this is, you are an expert in it, and the editors respect you as such. So, if you hate the paper, and think it should not be published, I don't see the downside to telling this to all editors who ask. If you don't hate the paper, but think it is not quite Annals-worthy, you can just say in your review "I don't think this paper is suitable for Annals of Math, but I think it would be well-placed in the Albanian Journal of Irreproducible Results (or whatever)." Once you have an internally consistent view on the paper, again, I see no problem with communicating this to editors, and you are doing the author a favor, since the second review is very quick.
In fact, in many cases, editors work on multiple journals, and I have heard of papers being submitted to (eg) Annals, and then being accepted to (eg) the Albanian Journal of IR. Presumably, this makes the author reasonably happy, since the paper actually appears somewhere in finite time.
A: I have had similar experiences to the OP and I think the OP is handling it correctly. If I'm rejecting a paper which is (apparently) correct and new on the basis of an opinion that the result is not good enough for journal X and I get it again from journal Y of a similar quality as X, then clearly I have a different opinion from the author about the paper's quality and I think the author deserves a second opinion. 
A: I agree with Felipe.  When a mathematician resubmits a paper to a different journal, she is soliciting a second opinion, as I think is her right.  If there was a clear, serious error that you found the first time you refereed a paper and you still see it in the second version, then I think it would be helpful to communicate that to the editor.  But I would guess that's a relatively rare situation: most papers are rejected because the referee just doesn't feel they are up to the journal's snuff.  Get someone else to decide this the second time.  
A: Note throughout that there are two only-partly related issues: literal correctness, and "importance", and that the latter is tangled up with "status" and "relative prestige" of journals. And, don't forget, all journals get far more excellent papers than they can fit into their "pages", so they will reject many excellent ones, for essentially random reasons... because, looked at bluntly, the job of an editor is to reject papers (not to accept).
It is also the job of editors to maintain or enhance the reputation of their journal, while the real reason mathematicians "need" to publish is to maintain or enhance their reputation, and there's a delicate dance done to see who benefits and who "sacrifices", reputation-wise. In that context, I'd tend to bet that confessing prior rejection wouldn't help anything at all, since it resembles telling someone that you didn't really want to go with them to the prom, but you'd already asked other people and were refused.
In that context, I think one should decline to referee a paper a second time for a different journal, if only on the principle that it might be that one gives a negative opinion for (accidentally) subjective reasons, e.g., that the author's priorities are not what they ought to be. One may truly believe this, but declaring that the author should have written a different paper entirely is a hard objection to meet. If the objections are arguably "objective", that there're serious tangible errors, mistatements of fact, disregard of prior art, etc., that's of course a different matter, but I think these pseudo-objective concerns are not the usual watershed for publication-or-not. It's status/reputation.
