We model explosions driven by the coalescence of a black hole or neutron star with the core of its massive-star companion. Upon entering a common-envelope phase, a compact object may spiral all the way to the core. The concurrent release of energy is likely to be deposited into the surrounding common envelope, powering a merger-driven explosion. We use hydrodynamic models of binary coalescence to model the common-envelope density distribution at the time of coalescence. We find toroidal profiles of material, concentrated in the binary's equatorial plane and extending to many times the massive star's original radius. We use the spherically averaged properties of this circumstellar material (CSM) to estimate the emergent light curves that result from the interaction between the blast wave and the CSM. We find that typical merger-driven explosions are brightened by up to three magnitudes by CSM interaction. From population synthesis models, we discover that the brightest merger-driven explosions, M-V similar to -18 to -20, are those involving black holes because they have the most massive and extended CSM. Black hole coalescence events are also common; they represent about 50% of all merger-driven explosions and approximately 0.25% of the core-collapse rate. Merger-driven explosions offer a window into the highly uncertain physics of common-envelope interactions in binary systems by probing the properties of systems that merge rather than eject their envelopes.