In this paper, we explore a multi-task semantic communication (SemCom) system for distributed sources, extending the existing focus on collaborative single-task execution. We build on the cooperative multi-task processing introduced in [1], which divides the encoder into a common unit (CU) and multiple specific units (SUs). While earlier studies in multi-task SemCom focused on full observation settings, our research explores a more realistic case where only distributed partial observations are available, such as in a production line monitored by multiple sensing nodes. To address this, we propose an SemCom system that supports multi-task processing through cooperation on the transmitter side via split structure and collaboration on the receiver side. We have used an information-theoretic perspective with variational approximations for our end-to-end data-driven approach. Simulation results demonstrate that the proposed cooperative and collaborative multi-task (CCMT) SemCom system significantly improves task execution accuracy, particularly in complex datasets, if the noise introduced from the communication channel is not limiting the task performance too much. Our findings contribute to a more general SemCom framework capable of handling distributed sources and multiple tasks simultaneously, advancing the applicability of SemCom systems in real-world scenarios.