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Intelligent reflecting surfaces (IRSs)-assisted wireless communication promises improved system performance, while posing new challenges in channel estimation (CE) due to the passive nature of the reflecting elements. Although a few CE protocols for IRS-assisted multiple-input single-output (MISO) systems have appeared, they either require long channel training times or are developed under channel sparsity assumptions. Moreover, existing works focus on a single IRS, whereas in practice multiple such surfaces should be installed to truly benefit from the concept of reconfiguring propagation environments. In light of these challenges, this paper tackles the CE problem for the distributed IRSs-assisted multi-user MISO system. An optimal CE protocol requiring relatively low training overhead is developed using Bayesian techniques under the practical assumption that the BS-IRSs channels are dominated by the line-of-sight (LoS) components. An optimal solution for the phase shifts vectors required at all IRSs during CE is determined and the minimum mean square error (MMSE) estimates of the BSusers direct channels and the IRSs-users channels are derived. Simulation results corroborate the normalized MSE (NMSE) analysis and establish the advantage of the proposed protocol as compared to benchmark scheme in terms of training overhead.