You should be able to use SharePoint's draft item security here.
Within the document library settings, select versioning settings then select the following:
- Require content approval for submitted items - yes
- Create major and minor (draft) versions
- Under draft item security select 'only users can approve items (and the author of the item).
Ensure that the site owners are part of the approvers group and they should then be able to see all items in the library (draft or published), whereas normal users who are not part of the approvers group can only see their own items.
Log in as both a normal user and a site owner to check that what you can see is correct.