We consider a decentralized multi-sensor estimation problem where L sensor nodes observe noisy versions of a correlated random source vector. The sensors amplify and forward their observations over a fading coherent multiple access channel (MAC) to a fusion center (FC). The FC is equipped with a large array of N antennas, and adopts a minimum mean square error (MMSE) approach for estimating the source. We optimize the amplification factor (or equivalently transmission power) at each sensor node in two different scenarios: a) with the objective of total power minimization subject to mean square error (MSE) of source estimation constraint, and b) with the objective of minimizing MSE subject to total power constraint. For this purpose, based on the well-known favorable propagation condition (when L ≪ N) achieved in massive multiple-input multiple-output (MIMO), we apply an asymptotic approximation on the MSE, and use convex optimization techniques to solve for the optimal sensor power allocation in a) and b). In a), we show that the total power consumption at the sensors decays as 1/N, replicating the power savings obtained in massive MIMO mobile communications literature. We also show several extensions of the aforementioned scenarios to the cases where sensor-to-FC fading channels are correlated, and channel coefficients are subject to estimation error. Through numerical studies, we also illustrate the superiority of the proposed optimal power allocation methods over uniform power allocation.