We propose an extension of the classical Rényi divergences to quantum states through an optimization over probability distributions induced by restricted sets of measurements. In particular, we define the notion of locally-measured Rényi divergences, where the set of allowed measurements originates from locality constraints between (distant) parties A and B. As our main result, we derive variational characterizations of these locally-measured Rényi divergences. We then evaluate them for variants of data-hiding states, showcasing the reduced distinguishing power of locality-constrained measurements, and give corresponding applications in locally-measured hypothesis testing.