We introduce a general model for the local obfuscation of probability distributions by probabilistic perturbation, e.g., by adding differentially private noise, and investigate its theoretical properties. Specifically, we relax a notion of distribution privacy (DistP) by generalizing it to divergence, and propose local obfuscation mechanisms that provide divergence distribution privacy. To provide f-divergence distribution privacy, we prove that probabilistic perturbation noise should be added proportionally to the Earth mover's distance between the probability distributions that we want to make indistinguishable. Furthermore, we introduce a local obfuscation mechanism, which we call a coupling mechanism, that provides divergence distribution privacy while optimizing the utility of obfuscated data by using exact/approximate auxiliary information on the input distributions we want to protect.