This paper proposes a distributed coordination scheme for connected vehicles, including automated vehicles (AVs) and manual vehicles (MVs), at signal-free intersections. The cooperation issue of vehicles at an intersection is formulated into a multi-objective optimization problem that aims to eliminate conflicts and improve traffic mobility and fuel economy. For this purpose, the future trajectories of AVs and MVs are predicted by the respective car-following models, and are shared with neighboring vehicles in conflict relationships. The proposed scheme optimizes the sum of the performance of AVs within the cooperative zone in a prediction horizon. A distributed optimization algorithm in the receding horizon is presented to obtain the local optimal solutions, and is tested in simulations with different demand levels and penetration rates of AVs. The results show that the proposed scheme reduces travel time by 29.7–45.5% and 34.5–49.2%, and decreases fuel consumption by 27.6–35.3% and 21.6–29.9% under 70–100% penetration rates of AVs, compared to the no-control operation and fixed-time signal control strategy. In addition, a comparison simulation with the strategy of jointly optimizing the vehicle trajectory and signal timing is conducted to evaluate the relative merits of the proposed scheme.