Scientific papers, like all other sorts of writing or creative expression, are covered by copyright. And, this is potentially a very bad thing.
Copyright grants a lot of sweeping rights to the writers of a paper, or the producers of any creative work. Often, those rights get signed away to a publisher or a distributor, but they are still there and enforceable by law. Among those rights is the right the term "copyright" is named after: the right to make copies of the work, but more significantly, the right to prevent anybody else from making a copy. There's another right that goes even further, and that some are not aware of: the right to make "derivative works." You cannot legally create and distribute a movie or novel about Luke Skywalker and Chewbacca, because 20th Century Fox (or Lucasfilms, or somebody) owns the copyright to the works in which those characters were created, and you need a license from them to distribute derivative works.
Scientists do not need, and indeed should not have, exclusive (or any) control over who can copy their papers, and who can make derivative works of their papers.
The very progress of science is based on derivative works! It is absolutely essential that somebody else who attempts to reproduce your experiment be able to publish results that you don't like if those are the results they have. Standard copyright, however, gives the copyright holders of a paper at least a plausible legal basis on which to challenge the publication of a paper that attempts to reproduce the results— clearly a derivative work!
There is a sort of community agreement in science that we do not assert the legal powers given to us by copyright, because we all understand that the progress of science is dependent on derivative works. However, the recent affair with Shelly Batts' blog and Wiley (some of Shelly's posts are here, here, and here) shows that we are living on tenuous ground. Not only can individual scientists use copyright to legally challenge the progress of science, but too often we have signed copyrights over to large corporate entities who may become overzealous in protecting "their rights," at the expense of the open discourse that is essential to the progress of science.
Because derivative works and direct copying sometimes become murky, the fact neither scientists nor journals or anybody else should have the power to prevent derivative works means that they also should not have the power to prevent copying of scientific papers. But there is another reason: the very reason that a society supports basic scientific research is at odds with the assumptions behind copyright. (There is a difference between the sort of "R&D" that a for-profit entity might do and basic research. And, yes, while some is clearly on one side of the line, and some (like astronomy) is on the other, there is a big grey area that makes it hard to draw the line. I'm not going to get into that right now.) Basic scientific research is not profitable in the same way that R&D, creation of teaching materials, or creation of entertainment is. As such, copyright as a means for creators to produce revenue, thereby as an incentive for creation, doesn't completely make sense. How is basic science funded? Partially through private foundations and charitable gifts, but largely through government (i.e. taxpayer) funding. Society as a whole pays for what we do, because society as a whole recognizes the value of focused curiosity and exploration. Given that, however, it is completely inappropriate for the papers and results that we produce directly from that exploration to be protected behind a copyright wall. Everybody should be able to freely copy the papers we write, because by and large they have already paid for it! Because by and large the only justification there is for our being paid to do it is that society at large is interested in what we're finding out!
(Textbooks are another matter; I will point you to this letter to AER that I wrote pondering that issue, and not write more about it now.)
A commenter on a previous article put forward this straw man: if you're opposed to copyrights on scientific papers, then perhaps we should just all start publishing scientific results anonymously, and only worry about the pure science. This is a straw man because there is a huge gulf between rejecting exclusive control over copying and derivative works, and not receiving any credit for what you've done.
We need attribution in science for two reasons. The first reason is that that is how our performance is judged. We absolutely need to have our names on our papers so that the universities, labs, and government funding agencies that pay us and provide us with research funds have some way of judging how much we're doing. Similarly, when another scientist uses are work, we need them to cite our work.
The second reason is for the reader. When I go and look at a list of scientific papers, I look at the author list. If the paper looks like it might be a little nutty, but the author is somebody whose work I know and whom I respect, I'm going to realize that I have to at least give the paper a chance to be taken seriously. Scientists are people, some are better than others, and some do more respectable work than others. Who is behind a given paper is useful information.
In science, fully citing the works that you are basing your work on is considered a part of professional ethics. Indeed, the ethical requirements for citation go beyond what is required for copyright. You can legally write a time travel story without citing H.G. Wells, or whatever other author you got the idea from. Ethically in science, however, you should cite the paper that the ideas are based on. (This does not mean that Wells' work would always be cited, for eventually later works become the more concrete basis. It's complicated.)
The sort of copyright that we need is something like an "Attribution-Share Alike" Creative Commons license. We absolutely should not have, nor should journals have, any sort of exclusive right to prevent reuse of our papers. But we do need credit and citation. I wouldn't proposal any sorts of laws for this. Indeed, putting all scientific papers in the public domain would be fine. Leave the requirement for citation as a matter of professional ethics. Universities, labs, and scientific societies take such professional ethics very seriously, and that' s enough enforcement; we don't need laws. As the Batts/Wiley affair shows, when laws get involved, sometimes very much the wrong interests can flex their muscles in ways that are inappropriate to science.