We consider personalized estimation for heterogeneous data under communication constraints. In many applications, distributed users have heterogeneous local data with distinct statistics, and want to estimate individual (personalized) properties of the local data. However, they have limited local data and we explore how collaboration (even over communication-limited links) can enable better personalized estimation. We study this for the Gaussian Bayesian model for heterogeneity with unknown parameters and a worst-case total regret criterion. We characterize (order-wise) the worst-case regret for personalized mean estimation by devising novel lower bounds and achievability schemes, which also demonstrates the value of collaboration.