Air conditioning systems are promising energy flexibility resources for smart grids. However, buildings with various thermodynamics must be coordinated to utilize limited energy flexibility effectively. This study proposes a distributed coordination strategy to coordinate building flexible thermal loads of different characteristics for optimized utilization of energy flexibility in a scalable and distributed manner. It consists of two components:1) an average consensus-based distributed sensing scheme to estimate the average thermal state of charge (SoC) of multiple zones, and 2) a weighted consensus-based distributed allocation module to allocate the demand response (DR) tasks or limited energy resources to multiple zones, proportional to their thermal storage capacities and deviations to the average thermal SoCs. Both algorithms achieve their goals respectively by fully distributed means through a sparse network with neighbor-to-neighbor communication. The sufficient condition for converging the weighted consensus algorithm is also derived for the first time. The proposed strategy is adopted for 1) weighted DR participation of residential inverter air conditioners and 2) weighted water flow redistribution of the commercial building water heating systems under urgent DR events. Simulation results show that adopting the distributed coordination strategy avoids the early depletion of demand flexibility resources and nonuniform thermal comfort sacrifices under uncoordinated control.