Autonomous drone swarms deployed for surveillance, environmental monitoring, and infrastructure inspection must maintain reliable coverage of critical assets despite robot failures. This requires multi-coverage: each asset must be observed by multiple robots for redundancy, with coverage requirements varying by asset importance. While recent work [3] has solved the centralized problem optimally using integer programming, practical deployments face constraints that demand distributed solutions: robots operate with limited communication ranges, onboard computation restricts global planning, and partial system failures must not cause mission abort. We present a distributed multi-coverage algorithm for robot swarms operating with local sensing, local communication, and no global coordination (Code available at: https://doi.org/10.5281/zenodo.18626854 ).