The Global Network of Optical Magnetometers to search for Exotic physics (GNOME) is a network of geographically separated, time-synchronized, optically pumped atomic magnetometers that is being used to search for correlated transient signals heralding exotic physics. The GNOME is sensitive to nuclear- and electron-spin couplings to exotic fields from astrophysical sources such as compact dark-matter objects (for example, axion stars and domain walls). Properties of the GNOME sensors such as sensitivity, bandwidth, and noise characteristics are studied in the present work, and features of the networks operation (e.g., data acquisition, format, storage, and diagnostics) are described. Characterization of the GNOME is a key prerequisite to searches for and identification of exotic physics signatures.