In my research, I've found two articles that are completely identical, except the abstract and authors names. They are published on different conferences, with one year between.
What should (must) I do in this case?
The programme committees for the conferences would be the appropriate authorities to contact (it is their responsibility to deal with plagiarism within their conference). This is of course complicated by the fact the conferences are in the past; the current programme committee may have little or nothing to do with the earlier committee, the steering committee may be more appropriate in this case.
If neither are responsive, contacting the publication venue may be required. If the proceedings are published by a journal or reputable publisher, they should be able to resolve the issue (they have copyright laws to deal with!).
If the author sets on the two papers are disjoint, then it looks like a case of real plagiarism. If you are not an author on any of these papers, I'd suggest notifying the authors of the earlier publication of the other paper, and leave further steps to them. Alternatively, if the conference of the earlier publication has been sponsored by a reputable organization, you could try to inform this organization.
If the author sets are not disjoint, we have what is typically called "self-plagiarism". I don't like this term, because this is by far not as severe as real plagiarism. Still, in case the earlier publication has a copyright mark by some publishing company or scientific organization, you may consider informing them.
Note that you are not really obliged to take any action. Just if you want to cite the work, you should make sure to cite the original, not the plagiated work. This would normally be the earlier publication.
Actually it sometimes occurs that essentially the same paper is published multiple times in the proceedings of different conferences sponsored by the same society and handled by the same publisher.
I personally disagree strongly with this but it is tolerated in some cases, on the grounds that proceedings to different conferences reach different audiences (especially if the events are well separated in time). In the case I'm familiar with, it was enough for the authors to indicate in a footnote to the 2nd paper that this work has already been presented at conference YYY. Obviously such duplications cannot involve two completely distinct sets of authors. (I've never heard this done "across publisher" as this could be problematic from a copyright perspective.)
You need to contact the publishers with this information, and let them deal with any follow up inquiry.