Visible-light integrated photonics is emerging as a promising technology for the realization of optical devices for applications in sensing, quantum information and communications, imaging, and displays. Among the existing photonic platforms, high-index-contrast silicon nitride (Si3N4) waveguides offer broadband transparency in the visible spectral range and a high scale of integration. As the complexity of photonic integrated circuits (PICs) increases, on-chip detectors are required to monitor their working point for reconfiguration and stabilization operations. In this Letter, we present a semi-transparent in-line power monitor integrated on Si3N4 waveguides that operates in the red-light wavelength range (660 nm). The proposed device exploits the photoconductivity of a hydrogenated amorphous-silicon (a-Si:H) film that is evanescently coupled to an optical waveguide. Experimental results show a responsivity of 30 mA/W, a sensitivity of -45 dBm, and a sub-µs time response. These features enable the use of the proposed photoconductor for high-sensitivity monitoring and control of visible-light Si3N4 PICs.