In this work, we study the problem of communication-constrained distributed estimation of the mean of a random variable drawn from a location family. We assume that each user has an independent and identically distributed sample drawn from a continuous distribution. Each user is allowed to communicate a limited number of bits to a central server, who must estimate the true mean of the distribution with low mean squared error. Recently, an order-optimal solution was proposed for the specific case of the Gaussian distribution. We propose a similar protocol for this problem, and derive upper bounds on the achievable mean squared error for symmetric log-concave distributions. © 2023 IEEE.