Semantic communication shifts the focus from bit-level accuracy to task-relevant semantic delivery, enabling efficient and intelligent communication for next-generation networks. However, existing multi-modal solutions often process all available data modalities indiscriminately, ignoring that their contributions to downstream tasks are often unequal. This not only leads to severe resource inefficiency but also degrades task inference performance due to irrelevant or redundant information. To tackle this issue, we propose a novel task-aware distributed information bottleneck (TADIB) framework, which quantifies the contribution of any set of modalities to given tasks. Based on this theoretical framework, we design a practical coding scheme that intelligently selects and compresses only the most task-relevant modalities at the transmitter. To find the optimal selection and the codecs in the network, we adopt the probabilistic relaxation of discrete selection, enabling distributed encoders to make coordinated decisions with score function estimation and common randomness. Extensive experiments on public datasets demonstrate that our solution matches or surpasses the inference quality of full-modal baselines while significantly reducing communication and computational costs.