We present the Local Self-Balancing sampler (LSB), a local Markov Chain Monte Carlo (MCMC) method for sampling in purely discrete domains, which is able to autonomously adapt to the target distribution and to reduce the number of target evaluations required to converge. LSB is based on (i) a parametrization of locally balanced proposals, (ii) an objective function based on mutual information and (iii) a self-balancing learning procedure, which minimises the proposed objective to update the proposal parameters. Experiments on energy-based models and Markov networks show that LSB converges using a smaller number of queries to the oracle distribution compared to recent local MCMC samplers.