Federated learning enables training machine learning models while preserving the privacy of participants. Surprisingly, there is no differentially private distributed method for smooth, non-convex optimization problems. The reason is that standard privacy techniques require bounding the participants' contributions, usually enforced via $\textit{clipping}$ of the updates. Existing literature typically ignores the effect of clipping by assuming the boundedness of gradient norms or analyzes distributed algorithms with clipping but ignores DP constraints. In this work, we study an alternative approach via $\textit{smoothed normalization}$ of the updates motivated by its favorable performance in the single-node setting. By integrating smoothed normalization with an error-feedback mechanism, we design a new distributed algorithm $α$-$\sf NormEC$. We prove that our method achieves a superior convergence rate over prior works. By extending $α$-$\sf NormEC$ to the DP setting, we obtain the first differentially private distributed optimization algorithm with provable convergence guarantees. Finally, our empirical results from neural network training indicate robust convergence of $α$-$\sf NormEC$ across different parameter settings.