We consider in this paper a new intelligent reflecting surface (IRS)-aided LEO satellite communication system, by utilizing the controllable phase shifts of massive passive reflecting elements to achieve flexible beamforming, which copes with the time-varying channel between the high-mobility satellite (SAT) and ground node (GN) cost-effectively. In particular, we propose a new architecture for IRS-aided LEO satellite communication where IRSs are deployed at both sides of the SAT and GN, and study their cooperative passive beamforming (CPB) design over line-of-sight (LoS)-dominant single-reflection and double-reflection channels. Specifically, we jointly optimize the active transmit/receive beamforming at the SAT/GN as well as the CPB at two-sided IRSs to maximize the overall channel gain from the SAT to each GN. Interestingly, we show that under LoS channel conditions, the high-dimensional SAT-GN channel can be decomposed into the outer product of two low-dimensional vectors. By exploiting the decomposed SAT-GN channel, we decouple the original beamforming optimization problem into two simpler subproblems corresponding to the SAT and GN sides, respectively, which are both solved in closed-form. Furthermore, we propose an efficient transmission protocol to conduct channel estimation and beam tracking, which only requires independent processing of the SAT and GN in a distributed manner, thus substantially reducing the implementation complexity. Simulation results validate the performance advantages of the proposed IRS-aided LEO satellite communication system with two-sided cooperative IRSs, as compared to various baseline schemes such as the conventional reflect-array and one-sided IRS.