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In cell-free massive multiple-input multiple-output systems, a large number of distributed wireless access points (AP) simultaneously serve a number of user equipments (UEs). This setup has recently been introduced as a promising alternative for the current 5G cellular networks. The setup has the ability to offer a good quality of service, especially for the UEs on the cell edges, be it that there is still a need for low-complexity signal processing algorithms. In this paper, the problem of optimal power allocation combined with uplink receive combining and downlink transmit precoding is tackled by providing efficient distributed MMSE-based algorithms. The necessary fronthaul communications to estimate the combining/precoding vectors and the necessary large-scale channel statistics are reduced to a minimum and rely on in-network summation that can be accomplished whenever the APs can be arranged into a tree-topology. Non-weighted max-sum and max–min are used as utilities for the power allocation, but the algorithms are not limited to these cases. Simulations show that the proposed algorithms outperform heuristic power allocation methods, both in uplink and downlink. [ABSTRACT FROM AUTHOR]