Before revision:  Our main technical contribution is an approach to handle the correlation that builds due to interactivity and quantifies how effectively one can exploit this correlation in spite of the local constraints. Using this, we fill gaps in some prior works and characterize the optimal sample complexity of learning and testing discrete distributions under total variation distance, for both communication and local differential privacy constraints. Prior to our work, this was known only for the problem of testing under local privacy constraints (Amin, Joseph, and Mao (2020);
After revision:  Our main technical contribution is an approach to handle the correlation that builds due to interactivity ; in fact, gaps can be found in prior claims of tight bounds of distribution estimation using interactive protocols. We propose a new approach to handle this correlation and establish a unified method to establish lower bounds for both tasks. As an application, we obtain optimal bounds for both estimation and testing under local privacy constraints (Amin, Joseph, and Mao (2020);