In this paper, the problem of stochastic synchronization analysis is investigated for two bidirectionally coupled piecewise linear chaotic systems disturbed by the common or different Poisson noise. By using the key tool such as the infinitesimal operator for stochastic differential equations driven by the Poisson process. Rigorous conditions are established for achieving complete synchronization with probability 1. We also find that with the common noise it is easier to induce the synchronization than with different noise. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed approach.