Cooperative control of connected and automated vehicles (CAVs) promises smoother traffic flow. In mixed traffic, where human-driven vehicles with unknown dynamics coexist, datadriven predictive control techniques allow for CAV safe and optimal control with measurable traffic data. However, the centralized control setting in most existing strategies limits their scalability for large-scale mixed traffic flow. To address this problem, this paper proposes a cooperative DeeP-LCC (Data-EnablEd Predictive Leading Cruise Control) formulation and its distributed implementation algorithm. In cooperative DeeP-LCC, the traffic system is naturally partitioned into multiple subsystems with one single CAV, which collects local trajectory data for subsystem behavior predictions based on the Willems' fundamental lemma. Meanwhile, the cross-subsystem interaction is formulated as a coupling constraint. Then, we employ the Alternating Direction Method of Multipliers (ADMM) to design the distributed DeeP-LCC algorithm. This algorithm achieves both computation and communication efficiency, as well as trajectory data privacy, through parallel calculation. Our simulations on different traffic scales verify the real-time wave-dampening potential of distributed DeeP-LCC, which can reduce fuel consumption by over 31.84% in a large-scale traffic system of 100 vehicles with only 5%-20% CAVs.