This paper advocates the use of the distributed compressed sensing (DCS)paradigm to deploy energy harvesting (EH)Internet of Thing (IoT) devices for energy self-sustainability. We consider networks with signal/energy models that capture the fact that both the collected signals and the harvested energy of different devices can exhibit cor-relation. We provide theoretical analysis on the performance of both the classical compres-sive sensing (CS) approach and the proposed distributed CS (DCS)-based approach to data acquisition for EH IoT. Moreover, we perform an in-depth comparison of the proposed DCS-based approach against the distributed source coding (DSC) system. These performance characterizations and comparisons embody the effect of various system phenomena and pa-rameters including signal correlation, EH cor-relation, network size, and energy availability level. Our results unveil that, the proposed ap-proach offers significant increase in data gath-ering capability with respect to the CS-based approach, and offers a substantial reduction of the mean-squared error distortion with respect to the DSC system.