Cloud storage systems must process massive data access requests with varying user privileges. Traditional ABE access control lacks granularity and flexibility for complex security needs, while Key Generation Centers (KGC) pose reliability risks and potential key leakage vulnerabilities. Therefore, we propose a Weakly Centralized Hierarchical Sensitive Data Sharing (WCHSDS) scheme based on edge computing. Firstly, the scheme integrates user attributes with diverse access permissions into a hierarchical access control tree, allowing for fine-grained control over user privileges. Secondly, considering the security risks posed by the Key Generation Center (KGC), this scheme introduces a weakly centralized key management strategy where users retain partial private keys, thereby eliminating complete dependence on the KGC. Lastly, to address efficiency challenges in edge computing scenarios, the WCHSDS offloads partial data search and attribute decryption tasks to edge servers, significantly improving overall system performance. Performance and security analysis demonstrate that the WCHSDS scheme achieves the designed privacy, security, and efficiency objectives. The experimental results demonstrate that the edge computing approach reduces computational overhead for users.