One of the most important goals of cooperative driving is to control connected automated vehicles (CAVs) passing through conflict areas safely and efficiently without traffic signals. As a typical application scenario, allocating right-of-way reasonably at unsignalized intersections can effectively avoid collisions and reduce traffic delays. Proposed here is a new cooperative driving strategy for CAVs at unsignalized intersections based on distributed Monte Carlo tree search (MCTS). A task-area partition framework is also proposed to decompose the mission of cooperative driving into three main tasks: vehicle information sharing, passing order optimization, and trajectory control. Based on the schedule tree of the vehicle passing order, the root parallelization of MCTS combined with the majority voting rule is used to explore as many feasible passing orders (leaf nodes) as possible in a distributed way and find a nearly global-optimal passing order within the limited planning time. The aim is for CAVs to perform proper trajectory adjustments based on the obtained passing order to minimize traffic delays while making the slightest acceleration adjustments. A coupled simulation platform integrating SUMO and Python is developed to construct the unsignalized intersection scenarios and generate the proposed distributed cooperative driving strategy. Comparative analysis with conventional driving strategies demonstrates that the proposed strategy significantly enhances efficiency, safety, comfort, and emission, aligning well with innovative and environmentally friendly urban mobility aspirations.