I know for international productions they are often given the option to buy a production's design, logos, etc. Sometimes there is not even a choice, and these productions must be a replica of the original. In case creatives from the original are involved (they often send choreographers and directors involved in the original staging) they are sometimes given the freedom to "adapt" certain aspects of the show (costume or set pieces, etc.) to "freshen" it up.
So, I would say the producers own these designs, so they can exploit them to their will.
Listen, I don't take my clothes off for anyone, even if it is "artistic". - JANICE