In this paper, we consider Wiener filters to reconstruct deterministic and (wide-band) stationary graph signals from their observations corrupted by random noises, and we propose distributed algorithms to implement Wiener filters and inverse filters on networks in which agents are equipped with a data processing subsystem for limited data storage and computation power, and with a one-hop communication subsystem for direct data exchange only with their adjacent agents. The proposed distributed polynomial approximation algorithm is an exponential convergent quasi-Newton method based on Jacobi polynomial approximation and Chebyshev interpolation polynomial approximation to analytic functions on a cube. Our numerical simulations show that Wiener filtering procedure performs better on denoising (wide-band) stationary signals than the Tikhonov regularization approach does, and that the proposed polynomial approximation algorithms converge faster than the Chebyshev polynomial approximation algorithm and gradient decent algorithm do in the implementation of an inverse filtering procedure associated with a polynomial filter of commutative graph shifts.