We characterize the frequency noise performance of a free-running dual-comb source based on an erbium-doped glass chip running two adjacent mode-locked waveguide lasers. This compact laser platform, contained only in a 1.2 L volume, rejects common-mode environmental noise by 20 dB thanks to the proximity of the two laser cavities. Furthermore, it displays a remarkably low mutual frequency noise floor around 10 Hz2/Hz, which is enabled by its large-mode-area waveguides and low Kerr nonlinearity. As a result, it reaches a free-running mutual coherence time of 1 s since mode-resolved dual-comb spectra are generated even on this time scale. This design greatly simplifies dual-comb interferometers by enabling mode-resolved measurements without any phase lock.