This depends on the legal agreement between you and your users. When a user signs on to your service, is there a checkbox they click to agree to some terms of service? That's where it should be spelled out.
It's typical for a forum to "own" all user-generated content. But beware there is a difference between ownership, copyright, and authorship. Each have their own set of rights and responsibilities. These are usually spelled out in a forum's Terms of Service or legal disclaimers.
Many forums (and other content sharing sites) do not claim ownership of user-generated media; instead they claim a worldwide royalty-free non-exclusive license to use the content. For example, Flickr does not own your photos, but by sending your photos to them you grant them a license to "use" them, ie store them and show them to the Interweb.
For example, WebmasterWorld puts this line in their ToS:
When a message is placed in any forum system, you are granting a soft license to the site to use it.