We demonstrate that separating beamforming (i.e., downlink precoding and uplink combining) and channel estimation in multi-user MIMO wireless systems incurs no loss of optimality under general conditions that apply to a wide variety of models in the literature, including canonical reciprocity-based cellular and cell-free massive MIMO system models. Specifically, we provide conditions under which optimal processing in terms of ergodic achievable rates can be decomposed into minimum mean-square error (MMSE) channel estimation followed by MMSE beamforming, for both centralized and distributed architectures. Applications of our results are illustrated in terms of concrete examples and numerical simulations.