Biochemical reactions are often subject to a complex fluctuating environment, which means that the corresponding reaction rates may themselves be time-varying and stochastic. If the environmental noise is common to a population of downstream processes, then the resulting rate fluctuations will induce statistical correlations between them. In this paper we investigate how such correlations depend on the form of environmental noise by considering a simple birth–death process with dynamical disorder in the birth rate. In particular, we derive expressions for the second-order statistics of two birth–death processes evolving in the same noisy environment. We find that these statistics not only depend on the second-order statistics of the environment, but the full generator of the process describing it, thus providing useful information about the environment. We illustrate our theory by considering applications to stochastic gene transcription and cell sensing.