A two-stage logistics service supply chain (LSSC) composed of one logistics service integrator (LSI) and one functional logistics service provider (LSP) is considered in this paper. It is assumed that the logistics service demand function is influenced by the LSI’s service level and LSP’s greening level, which can measure the impact of service quality and the environment. Under the centralized decision, LSI Stackelberg Game, LSP Stackelberg Game and Vertical-Nash decision power structure, four kinds of LSSC game models are proposed. In addition, their optimal solutions are derived and compared. Then the impact of power structures on the optimal solutions and profits are analysed. It is found that the optimal solutions are significantly influenced by the power structure, except for the optimal retail price. The greening level, service level, market demand, and overall profit of the supply chain are all the highest under centralized decision-making, followed by Vertical-Nash decision-making. Furthermore, cooperation and competition can improve the level of environmental protection and logistics industry service, and also can effectively expand the market demand and maximize the overall profit of the LSSC. Under a Stackelberg game decision-making scenario, the power of decision-rights exerts a positive impact on the profits of the LSSC participants.