Researchers have the original copyright to their own papers (except in some business-like employments). So the author can always give open access to her own original paper (the preprint) on a website. She can also give open access to an accepted, peer reviewed version (a postprint), if she has not explicitly transferred the copyright to this version to a publisher. But the published copy-edited version (reprint or offprint) of a paper will usually be copyright to the magazine, journal or publisher.
If a researcher has transferred all copyrights to a publisher after the acceptance of a peer reviewed paper, as it is commonly done, she cannot give open access to the accepted version (postprint) or the published copy-edited version (reprint or offprint) in the archive without infringing copyrights. Some publishers do however permit open access to either postprints or reprints, and some permit giving access to colleagues at ones own institution.
In order to make more research freely available, authors can request that the right to online self-archiving is retained in any future transfers of copyrights, and research institutions and other actors can make a concerted effort to back up these requests. If the authors have already transferred copyrights, they may request the publisher that they get back the rights to e.g. deposit the postprint in an open access archive and put it on their own homepage. Some publishers have stated that they grant these rights as a standard procedure. We hope that the initiatives taken by Organic Eprints and other actors will spur the promotion open access by reducing the problems with copyrights.
Open or restricted access
Organic Eprints does not want to infringe copyrights. We encourage giving open access where possible, but we also offer the possibility to restrict access. Access can be restricted to registered users at Organic Eprints, which is a limited and known group of people. Furthermore, access can be restricted to only depositor and archive staff. (The latter includes the depositing user, archive administrator and editors, and - by special agreement - selected research evaluators for an organization.)
The RoMEO/SHERPA website provides a useful search interface about the copyright policies of a wide range of publishers. It is also possible to search on journal titles.