In this paper, we propose a linear precoding technique for weighted sum rate (WSR) maximization in multiple-input multiple-output interfering broadcast channels. In multicell environments, the WSR can be jointly maximized through centralized processing which causes a large amount of channel state information (CSI) exchange. In order to reduce the overhead associated with CSI, we focus on a distributed precoding scheme utilizing local CSI at each base station (BS). First, applying a high signal-to-interference-plus-noise ratio assumption, we decouple the WSR maximization problem into distributed problems. Then we solve this distributed WSR maximization problem for each BS by using a zero-gradient based algorithm which converges to a local maximum point. Unlike conventional distributed schemes which require additional information, our proposed scheme at each base station utilizes only the local CSI to compute its precoding matrices. Through the Monte-Carlo simulation, we show that our proposed algorithm exhibits the performance almost identical to the centralized scheme requiring the global CSI.