Definitely consult a lawyer, preferably one who deals with copyright and intellectual property on a regular basis. I know quite a bit about the general principles of copyright, but specific applications of the law are really arcane. And copyright law is significantly different depending on the jurisdiction; what’s true in the UK is not necessarily true in the US.
Some general principles about US copyright:
“Public domain” refers to works that are not under copyright at all. In the US, this generally means things published before 1923, US government documents, and works that the author/copyright holder has explicitly declared to be in the public domain, i.e., explicitly given permission to anyone and everyone to republish or do with what they will. It is unlikely that any of the material on the forum in question is public domain. Just because you put something on the internet in a public forum does not make it “public domain.”
In almost all cases, the owner of the copyright is the author, as Eliza states. But the author may or may not be allowed to grant permission to republish the material to others. That may reside with the original publisher, the forum owner. A lot depends on the terms of service of the forum.
By posting the material on the forum, the author is implicitly giving the forum owner the license to publish the material on that forum. The terms of service may also give the forum owner license to publish the material elsewhere, but this must be explicitly stated somewhere. For example, the terms of service of the discussion forums run by the last company I worked for included this:
For content you post or otherwise provide to [the company] via this site, you grant [the company] permission to (1) use, copy, distribute, transmit, publicly display, publicly perform, reproduce, edit, modify, translate and reformat your content, and (2) sublicense these rights, to the maximum extent permitted by applicable law. [The company] will not pay you for your content. [The company] may remove your content at any time. For each content posted, you represent that you have all rights necessary for you to make the grants in this section. You agree that your posted content is not confidential, and [The company] may disclose your content to any law enforcement authorities or to any third party in the event it has a good faith reason to believe it is necessary for purposes of ensuring your compliance with these Rules of User Conduct, and protecting the rights, property, and interests of [the company].
In this case, by posting material to the forum the author gives the forum owner the right to use the material in any way it sees fit, including granting permission to publish to third parties (sub-licensing). The author still owns the copyright though and can publish her own material elsewhere without going to the forum owner for permission. If the forum in question has a sub-license clause like this one in its terms of service, then all you need is the forum owner’s permission to republish material--it saves you from having to go to all the different authors for separate permissions.
Other forums may demand an exclusive license to publish the material. This would be unusual for internet forums and would usually be for a limited time. But this is more typical of book publishing, where the author grants the publisher an exclusive license, but that is limited by either time or ends when the book goes out of print, at which time the author can take it to another publisher or self-publish it. In the case the author technically owns the copyright during the exclusive period, but the publisher has sole discretion regarding permission to republish material.