This paper proposes strategies for distributed Wiener-based reconstruction of graph signals from subsampled measurements. Given a stationary signal on a graph, we fit a distributed autoregressive moving average graph filter to a Wiener graph frequency response and propose two reconstruction strategies: (i) reconstruction from a single temporal snapshot; (ii) recursive signal reconstruction from a stream of noisy measurements. For both strategies, a mean square error analysis is performed to highlight the role played by the filter response and the sampled nodes, and to propose a graph sampling strategy. Our findings are validated with numerical results, which illustrate the potential of the proposed algorithms for distributed reconstruction of graph signals.