Astrometry from the Gaia mission was recently used to discover the two nearest known stellar-mass black holes (BHs), Gaia BH1 and Gaia BH2. These objects are among the first stellar-mass BHs not discovered via X-rays or gravitational waves. Both systems contain ∼1 M stars in wide orbits (a ≈ 1.4 au, 4.96 au) around ∼9 M BHs, with both stars (solar-type main sequence star, red giant) well within their Roche lobes in Gaia BH1 and BH2, respectively. However, the BHs are still expected to accrete stellar winds, leading to potentially detectable X-ray or radio emission. Here, we report observations of both systems with the Chandra X-ray Observatory, the Very Large Array (for Gaia BH1) and MeerKAT (for Gaia BH2). We did not detect either system, leading to X-ray upper limits of LX < 9.4 × 1028 and LX < 4.0 × 1029 erg s−1 and radio upper limits of Lr < 1.6 × 1025 and Lr < 1.0 × 1026 erg s−1 for Gaia BH1 and BH2, respectively. For Gaia BH2, the non-detection implies that the accretion rate near the horizon is much lower than the Bondi rate, consistent with recent models for hot accretion flows. We discuss implications of these non-detections for broader BH searches, concluding that it is unlikely that isolated BHs will be detected via interstellar medium accretion in the near future. We also calculate evolutionary models for the binaries' future evolution using Modules for Experiments in Stellar Astrophysics, and find that Gaia BH1 will be visible as a symbiotic BH X-ray binary for 5–50 Myr. Since no symbiotic BH X-ray binaries are known, this implies either that fewer than ∼104 Gaia BH1-like binaries exist in the Milky Way, or that they are common but have evaded detection.