We introduce and define the concept of a stochastic pooling network (SPN), as a model for sensor systems where redundancy and two forms of 'noise'—lossy compression and randomness—interact in surprising ways. Our approach to analysing SPNs is information theoretic. We define an SPN as a network with multiple nodes that each produce noisy and compressed measurements of the same information. An SPN must combine all these measurements into a single further compressed network output, in a way dictated solely by naturally occurring physical properties—i.e. pooling—and yet cause no (or negligible) reduction in mutual information. This means that SPNs exhibit redundancy reduction as an emergent property of pooling. The SPN concept is applicable to examples in biological neural coding, nanoelectronics, distributed sensor networks, digital beamforming arrays, image processing, multiaccess communication networks and social networks. In most cases the randomness is assumed to be unavoidably present rather than deliberately introduced. We illustrate the central properties of SPNs for several case studies, where pooling occurs by summation, including nodes that are noisy scalar quantizers, and nodes with conditionally Poisson statistics. Other emergent properties of SPNs and some unsolved problems are also briefly discussed.