Satellite-terrestrial integrated networks (STINs) are promising architecture for providing global coverage. In STINs, full frequency reuse between a satellite and a terrestrial base station (BS) is encouraged for aggressive spectrum reuse, which induces non-negligible amount of interference. To address the interference management problem in STINs, this paper proposes a novel distributed precoding method. Key features of our method are: i) a rate-splitting (RS) strategy is incorporated for efficient interference management and ii) the precoders are designed in a distributed way without sharing channel state information between a satellite and a terrestrial BS. Specifically, to design the precoders in a distributed fashion, we put forth a spectral efficiency decoupling technique, that disentangles the total spectral efficiency function into two distinct terms, each of which is dependent solely on the satellite's precoder and the terrestrial BS's precoder, respectively. Then, to resolve the non-smoothness raised by the RS strategy, we approximate the spectral efficiency expression as a smooth function by using the LogSumExp technique; thereafter we develop a generalized power iteration inspired optimization algorithm built based on the first-order optimality condition. Simulation results demonstrate that the proposed method offers considerable spectral efficiency gains compared to the existing methods.