The onerous media sharing task prompts resource-constrained media owners to seek help from a cloud platform, i.e., storing media contents in the cloud and letting the cloud do the sharing. There are three key security/privacy problems that need to be solved in the cloud media sharing scenario, including data privacy leakage and access control in the cloud, infringement on the owner's copyright, and infringement on the user's rights. In view of the fact that no single technique can solve the above three problems simultaneously, two cloud media sharing schemes are proposed in this paper, named FairCMS-I and FairCMS-II. By cleverly utilizing the proxy re-encryption technique and the asymmetric fingerprinting technique, FairCMS-I and FairCMS-II solve the above three problems with different privacy/efficiency trade-offs. Among them, FairCMS-I focuses more on cloud-side efficiency while FairCMS-II focuses more on the security of the media content, which provides owners with flexibility of choice. In addition, FairCMS-I and FairCMS-II also have advantages over existing cloud media sharing efforts in terms of optional IND-CPA (indistinguishability under chosen-plaintext attack) security and high cloud-side efficiency, as well as exemption from needing a trusted third party. Furthermore, FairCMS-I and FairCMS-II allow owners to reap significant local resource savings and thus can be seen as the privacy-preserving outsourcing of asymmetric fingerprinting. Finally, the feasibility and efficiency of FairCMS-I and FairCMS-II are demonstrated by experiments.