Semantic communication (SemCom) shifts the focus from data transmission to meaning delivery, enabling efficient and intelligent communication. Existing AI-based coding schemes for multi-modal multi-task SemCom often require transmitters with full-modal data to participate in all receivers' tasks, which leads to redundant transmissions and conflicts with the physical limits of channel capacity and computational capability. In this paper, we propose PoM$^2$-DIB, a novel framework that extends the distributed information bottleneck (DIB) theory to address this problem. Unlike the typical DIB, this framework introduces modality selection as an additional key design variable, enabling a more flexible tradeoff between communication rate and inference quality. This extension selects only the most relevant modalities for task participation, adhering to the physical constraints, while following efficient DIB-based coding. To optimize selection and coding end-to-end, we relax modality selection into a probabilistic form, allowing the use of score function estimation with common randomness to enable optimizable coordinated decisions across distributed devices. Experimental results on public datasets verify that PoM$^2$-DIB achieves high inference quality compared to full-participation baselines in various tasks under physical limits.

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