We investigate the radio and gamma-ray beaming properties of normal and millisecond pulsars by selecting two samples from the known populations. The first, Sample G, contains pulsars which are detectable in blind searches of gamma-ray data from the Fermi Large Area Telescope. The second, Sample R, contains pulsars detectable in blind radio searches which have spin-down luminosities Edot > 10^{34} erg/s. We analyse the fraction of the gamma-ray-selected Sample G which have detectable radio pulses and the fraction of the radio-selected Sample R which have detectable gamma-ray pulses. Twenty of our 35 Sample G pulsars have already observed radio pulses. This rules out low-altitude polar-cap beaming models if, as is currently believed, gamma-ray beams are generated in the outer magnetosphere and are very wide. We further find that, for the highest-Edot pulsars, the radio and gamma-ray beams have comparable beaming factors, i.e., the beams cover similar regions of the sky as the star rotates. For lower-Edot gamma-ray emitting pulsars, the radio beams have about half of the gamma-ray sky coverage. These results suggest that, for high-Edot young and millisecond pulsars, the radio emission originates in wide beams from regions high in the pulsar magnetosphere, probably close to the null-charge surface and to the gamma-ray emitting regions. Furthermore, it suggests that for these high-Edot pulsars, as in the gamma-ray case, features in the radio profile represent caustics in the emission beam pattern.