We propose distributed dual-comb spectroscopy, a quantum-enhanced sensing architecture that simultaneously exploits spatial- and spectral-domain entanglement to improve the extraction of global spectroscopic information across broadband frequency ranges. Our scheme consists of a network of Mach-Zehnder interferometers interrogating spatially separated samples, each driven by a classical frequency comb at the bright port. A single two-mode squeezed frequency comb is distributed through a passive beam-splitter array into K entangled combs, establishing multipartite correlations across sensors while preserving intrinsic spectral entanglement. The protocol is compatible with both heterodyne detection and division detection, and enables parallel estimation of weighted-average absorption features over multiple frequency components. Compared with spatially separable quantum strategies requiring K independently engineered nonclassical comb sources, our approach achieves enhanced signal-to-noise performance using only a single quantum resource distributed across the network. These results establish a scalable route toward broadband distributed quantum spectroscopy with simultaneous spatial and spectral advantage.