Sensor networks are important for monitoring several physical phenomena. In this paper, we consider the monitoring of diffusion fields and design simple, yet robust, sensing, data processing and communication strategies for estimating the sources of diffusion fields under communication constraints. Specifically, based on our previous work in the area, we firstly show how sources of the field can be recovered analytically through the use of well-chosen sensing functions. Then, by properly extending this scheme to our sensor network setting, we design and propose an effective diffusion field sensing strategy. Next, we introduce a physics-driven quantized gossip scheme, as a joint information processing and communication strategy for handling the network communication constraints: i.e. when a sensor can only communicate with a small subset of nodes over links with a finite capacity. Combining the proposed strategies allows us to develop a fully distributed algorithm for recovering sources of diffusion fields using sensor networks. Numerical simulation results are presented in order to evaluate the effectiveness and robustness of our algorithm.