Privacy-preserving sharing of cloud data requires user authentication and fine-grained access control along with privacy provisioning mechanisms. In this paper, we propose two schemes, keyword-based re-encryption (KRE) scheme and oblivious transfer-based re-encryption (OTRE) scheme for privacy-preserving secure health data sharing. KRE preserves data privacy and provides fine-grained access control with reduced key management overhead. OTRE is a superset of KRE that utilizes OT for users' data privacy and multiple data sharing. The schemes are collusion resistant against an honest but curious cloud and malicious users. The use of hybrid blockchain removes the need for a single trusted authority and provides large-scale secure data sharing. Blockchain also enables authentication between end-users and verifiability of data on the cloud through smart contracts. We implemented the schemes using the SageMath tool and hybrid blockchain using Go-Ethereum with Remix. The overall average time consumption of KRE scheme and OTRE scheme, 17:836ms and 23:62ms, respectively, for data sharing, indicates the efficiency of the schemes. The additional privacy provisioning step was responsible for the increased time in OTRE scheme. In addition to data sharing, consensus time, latency and throughput in KRE and OTRE schemes demonstrated their efficiency and suitability for healthcare data sharing applications.