At Admin Panel -> Settings -> Switches, the "allow claiming listings via edit/register" switch enables a system whereby any guest or member can ask to claim any unowned listing to their account. On the details page of unowned listings there'll be an "Is this your listing? Update and claim it!" link at the bottom which leads to the edit listing page. There, the person -- whether an unregistered guest or a member -- will be asked to write a message with evidence that they own the site, such as an email address on the listing's website domain that you can contact them at. They can then make any changes to the listing. If they're an unregistered guest, they'll be asked to choose a username and password and the username, password and email fields will all be marked as required for them. When they submit their changes, it'll be held as a pending listing edit.
As administrator, you'll review claims at Admin Panel -> Validation under the Listing Edits tab. If you're requiring validation of listing edits in other situations, you'll notice the ones that involve a claim of ownership have a special warning message that includes a link you can use to send them an email asking for clarification or further evidence if needed. Once you approve the pending edit, the claimant becomes the owner and can find their listing in their member home area.
Note that member registration must be enabled for this to work.