Localization within a known environment is a crucial capability for mobile robots. Simultaneous Localization and Mapping (SLAM) is a prominent solution to this problem. SLAM is a framework that consists of a diverse set of computational tasks ranging from real-time tracking to computation-intensive map optimization. This combination can present a challenge for resource-limited mobile robots. Previously, edge-assisted SLAM methods have demonstrated promising real-time execution capabilities by offloading heavy computations while performing real-time tracking onboard. However, the common approach of utilizing a client-server architecture for offloading is sensitive to server and network failures. In this article, we propose a novel edge-assisted SLAM framework capable of self-organizing fully distributed SLAM execution across a network of devices or functioning on a single device without connectivity. The architecture consists of three layers and is designed to be device-agnostic, resilient to network failures, and minimally invasive to the core SLAM system. We have implemented and demonstrated the framework for monocular ORB SLAM3 and evaluated it in both fully distributed and standalone SLAM configurations against the ORB SLAM3. The experiment results demonstrate that the proposed design matches the accuracy and resource utilization of the monolithic approach while enabling collaborative execution.