We study the distributed mean estimation (DME) problem under privacy and communication constraints in the local differential privacy (LDP) and multi-message shuffled (MMS) privacy frameworks. The DME has wide applications in both federated learning and analytics. We propose a communication-efficient and differentially private algorithm for DME of bounded $\ell _{2}$ -norm and $\ell _{\infty }$ -norm vectors. We analyze our proposed DME schemes showing that our algorithms have order-optimal privacy-communication-performance trade-offs. Our algorithms are designed by giving unequal privacy assignments at different resolutions of the vector (through binary expansion) and appropriately combining it with coordinate sampling. These results are directly applied to give guarantees on private federated learning algorithms. We also numerically evaluate the performance of our private DME algorithms.