This work focuses on the power minimization problem while ensuring target rates in the downlink of a multi-cell multi-user MIMO system wherein L base stations (BSs) of N antennas serve in total K single-antenna user equipments. We assume that the transmit antennas at each BS are correlated and propose a decentralized solution to compute an approximation of the optimal beamforming vectors. The analysis is conducted in the asymptotic regime in which N and K grow large with a given ratio K/N. In particular, the proposed solution relies on the exchange of intercell interference terms whose large system approximations are computed at each BS using knowledge of local transmit correlation matrices and non-local pathlosses. Numerical results are used to evaluate the performance loss of the proposed solution compared to the optimal one and to investigate its accuracy in systems of finite size.