We study node-specific speech enhancement in a distributed acoustic sensor network with one microphone per node, where each node retains its local talkers while suppressing competing speech, music, and background noise. We compare local-only and graph-conditioned models using matched backbone architectures under different communication regimes. Local processing already provides strong gains under favorable conditions, but degrades in low-SNR, high-interference scenarios. In these regimes, cross-node cooperation improves SI-SDR by up to 4.8~dB over the matched local model. We further characterize the performance--bandwidth tradeoff: low-rate bottleneck sharing provides moderate gains, whereas full feature sharing achieves the best performance at substantially higher communication cost.