Abstract: ‍ ‍Semantic communication， as an emerging communication paradigm， prioritizes the semantics of information over bit-level representation. It is characterized by high compression efficiency， effective transmission， and robust interference resistance. In the domain of semantic compression， this study introduces an innovative distributed source coding model tailored for multi-user scenarios， where each user independently extracts and compresses semantic information. Within this model， multiple users observe shared hidden semantic features， encode them independently， and perform joint decoding to reconstruct both the observed source and the semantic source under fidelity constraints. This paper presents a Berger-Tung rate-distortion theorem specific to the proposed model and extends the analysis to rate-distortion relationships under joint Gaussian source distributions. The analysis elucidates the trade-offs between the total rate of multiple users and the semantic and observation distortions， providing insights into how distortion constraints can be bidirectionally adjusted. In this context， the degradation of the generalized boundary in comparison to existing results is demonstrated， along with sufficient conditions for achieving inner and outer boundary matching. These sufficient conditions partially address the rate-distortion problem for joint Gaussian sources. The theoretical findings offer critical guidance for designing source coding in multi-user semantic communication systems.