In this paper we consider uniformly random lozenge tilings of arbitrary domains approximating (after suitable normalization) a closed, simply-connected subset of $\mathbb{R}^2$ with piecewise smooth, simple boundary. We show that the local statistics of this model around any point in the liquid region of its limit shape are given by the infinite-volume, translation-invariant, extremal Gibbs measure of the appropriate slope, thereby confirming a prediction of Cohn-Kenyon-Propp from 2001 in the case of lozenge tilings. Our proofs proceed by locally coupling a uniformly random lozenge tiling with a model of Bernoulli random walks conditioned to never intersect, whose convergence of local statistics has been recently understood by the work of Gorin-Petrov. Central to implementing this procedure is to establish a local law for the random tiling, which states that the associated height function is approximately linear on any mesoscopic scale.