Normally when you hide a cache you are listed as the cacher who placed the cache as well as the owner.
If you want to hide a cache with someone else, you must decide which of you will own the cache listing. That person will be those one whose caching name the cache is listed under and who will be able to edit the cache listing.
If you want, though, you can list both caching names in the Placed By field when you are hiding or editing a cache listing. This will make the cache appear to have been hidden by both of you.
Note that the cacher listing the cache will still retain ownership of the listing, the other party just gets kudos for helping in the hide.