I have no idea what the copyright laws say about this. There is a question I would like to reply to, and a figure that would make the reply more clear. However, the figure is from a PNAS paper, thus behind a paywall. I can access it. Does anybody know whether I am in principle allowed to reproduce the figure - i.e. make a jpeg out of it, upload it somewhere private and link it? I would reference it properly of course.
Anyone may, without requesting permission, use original figures or tables published in PNAS for noncommercial and educational use (i.e., in a review article, in a book that is not for sale) provided that the original source and the applicable copyright notice are cited.
Disclaimer: this is not professional legal advice and does not replace professional legal advice. I accept no liability for any use of this information.
So the implicit point made in @Chuck's answer is to check the copyright and permission section of a publisher's website. If the copyright holder grants permission, then it is fine (e.g., it would appear that including a figure is permitted by the APA).
Where such permission is not explicitly granted, then in some cases it might still be permissible under some other exception, such as the U.S. fair use exceptions. For example, in this review of fair use on the internet, the authors write:
In assessing whether a use of a copyrighted work is a “fair use,” courts weigh four statutory factors – (1) the purpose and character of the use, (2) the nature of the copyrighted work, (3) the amount and substantiality of the portion used in relation to the copyrighted work as a whole, and (4) the effect of the use upon the potential market for the copyrighted work. A bright line approach to fair use is difficult, if not impossible, to formulate, as courts examine the fair use defense on a case by case basis.
For example, the fact that Cogsci.SE seems to be largely for purposes of research and public dissemination of knowledge, the fact that a single figure is typically only a small percentage of the whole work, the fact that, if anything, presenting a figure is likely to actually act as advertising for the journal article rather than replace the journal article, support the fair use argument.
However, I'm not quite sure how this works on the Internet given that the content is available in many different jurisdictions presumably with slightly different copyright rules.