We study the distributed processing of graph signals that are well represented by graph spectral dictionaries. We first analyze the impact of quantization noise in the distributed computation of polynomial dictionary operators that are commonly used in various signal processing tasks. We show that the impact of quantization depends on the graph geometry and on the structure of the spectral dictionaries. Then, we focus on the problem of distributed sparse signal representation that can be solved with an iterative soft thresholding algorithm. We define conditions on the dictionary structure to ensure the convergence of the distributed algorithm and finally propose a dictionary learning solution that permits to control the robustness to quantization noise. Experimental results for reconstruction and denoising of both synthetic and practical signals illustrate the tradeoffs that exist between accurate signal representation and robustness to quantization error in the design of dictionaries operators in distributed graph signal processing.