We develop a novel framework to accelerate Gaussian process regression (GPR). In particular, we consider localization kernels at each data point to down-weigh the contributions from other data points that are far away, and we derive the GPR model stemming from the application of such localization operation. Through a set of experiments, we demonstrate the competitive performance of the proposed approach compared to full GPR, other localized models, and deep Gaussian processes. Crucially, these performances are obtained with considerable speedups compared to standard global GPR due to the sparsification effect of the Gram matrix induced by the localization operation.