Large-scale wireless sensor networks (WSNs) and Internet-of-Things (IoT) applications involve diverse sensing devices collecting and transmitting massive amounts of heterogeneous data. In this paper, we propose a novel compressive data aggregation and recovery mechanism that reduces the global communication cost without introducing computational overhead at the network nodes. Following the principles of compressive demixing, each node of the network collects measurement readings from multiple sources and mixes them with readings from other nodes into a single low-dimensional measurement vector, which is then relayed to other nodes; the constituent signals are recovered at the sink using convex optimization. Our design achieves significant reduction in the overall network data rates compared to prior schemes based on (distributed) compressed sensing or compressed sensing with (multiple) side information. Experiments using real large-scale air-quality data demonstrate the superior performance of the proposed framework against state-of-the-art solutions, with and without the presence of measurement and transmission noise.