Timed Data Release (TDR) is a practical security mechanism designed to safeguard data until a specified time has elapsed. Recently, a more general paradigm, referred to as Event-driven Data Release (EDR) has gained attention. EDR enables data to be released based on the occurrence of a specified event. Existing blockchain-based constructions for EDR either neglect privacy considerations or restrict themselves to specialized message spaces (e.g., signatures) while achieving provable privacy guarantees. Despite its practical relevance, constructing a privacy-preserving EDR framework that supports general message spaces on blockchain with formal security guarantees remains an open challenge. In this paper, we address this gap by introducing P-EDR, a privacy-preserving event-driven data release framework for general message spaces, built on smart contracts.P-EDR enables a data sender to encrypt its data, which can be securely released upon the occurrence of specified events. P-EDR incorporates a novel, tailored signature-based witness encryption, SGWE, designed to encrypt data messages within a general message space, contingent upon the presence of a valid signature of a referencing message. Using SGWE, decryption of ciphertext is only possible with a valid signature. We formally define the desired security model and present an efficient construction of SGWE. Additionally, we propose a practical design for P-EDR and we rigorously prove the security guarantees of our construction. Our experimental results for evaluating the performance of the proposed construction demonstrate the effectiveness of our approach.