We consider some general implications of bright gamma-ray counterparts to fast radio bursts (FRBs). We show that, even if these manifest in only a fraction of FRBs, gamma-ray detections with current satellites (including Swift) provide stringent constraints on cosmological FRB models. If the energy is drawn from the magnetic energy of a compact object such as a magnetized neutron star, the sources should be nearby and very rare. If the intergalactic medium is responsible for the observed dispersion measure, the required gamma-ray energy is comparable to that of the early afterglow or extended emission of short gamma-ray bursts. While this can be reconciled with the rotation energy of compact objects, as expected in many merger scenarios, the prompt outflow that yields the gamma-rays is too dense for radio waves to escape. Highly-relativistic winds launched in a precursor phase, and forming a wind bubble, may avoid the scattering and absorption limits and could yield FRB emission. Largely independent of source models, we show that detectable radio afterglow emission from gamma-ray bright FRBs can reasonably be anticipated. Gravitational wave searches can also be expected to provide useful diagnoses.