Speech dereverberation aims to alleviate the negative impact of late reverberant reflections. The weighted prediction error (WPE) method is a well-established technique known for its superior performance in dereverberation. However, in scenarios where microphone nodes are dispersed, the centralized approach of the WPE method requires aggregating all observations for inverse filtering, resulting in a significant computational burden. This paper introduces a distributed speech dereverberation method that emphasizes low computational complexity at each node. Specifically, we leverage the distributed adaptive node-specific signal estimation (DANSE) algorithm within the multichannel linear prediction (MCLP) process. This approach empowers each node to perform local operations with reduced complexity while achieving the global performance through inter-node cooperation. Experimental results validate the effectiveness of our proposed method, showcasing its ability to achieve efficient speech dereverberation in dispersed microphone node scenarios.