We research the issue of regionally non-public imply estimation of high-dimensional vectors within the Euclidean ball. Current algorithms for this drawback both incur sub-optimal error or have excessive communication and/or run-time complexity. We suggest a brand new algorithmic framework, ProjUnit, for personal imply estimation that yields algorithms which can be computationally environment friendly, have low communication complexity, and incur optimum error as much as a 1+o(1)-factor. Our framework is deceptively easy: every randomizer initiatives its enter to a random low-dimensional subspace, normalizes the outcome, after which runs an optimum algorithm akin to PrivUnitG within the lower-dimensional area. As well as, we present that, by appropriately correlating the random projection matrices throughout units, we are able to obtain quick server run-time. We mathematically analyze the error of the algorithm when it comes to properties of the random projections, and research two instantiations. Lastly, our experiments for personal imply estimation and personal federated studying reveal that our algorithms empirically get hold of almost the identical utility as optimum ones whereas having considerably decrease communication and computational price.