For rate optimization in interference limited network, improper Gaussian signaling has shown its capability to outperform the conventional proper Gaussian signaling. In this work, we study a weighted sum rate maximization problem with improper Gaussian signaling for the multiple-input multiple-output interference broadcast channel (MIMO-IBC). To solve this nonconvex and NP-hard problem, we propose an effective two-stage algorithm. In the first stage, a weighted mean square error (MSE) minimization algorithm is adopted to compute the transmit covariance matrices, and in the second stage, an alternating direction method of multipliers (ADMM)-based distributed multi-agent algorithm is proposed to obtain the globally optimal pseudo-covariance matrices in a parallel and scalable fashion. Finally, simulation results are presented to demonstrate the superior performance of our algorithm under different signal-to-noise ratio (SNR) and various network scenarios.