The principal discovery of the Compton Gamma-Ray Observatory (GRO) con
cerning gamma-ray bursts is that these sources are isotropic but with
a comparative deficiency of fainter sources, suggesting that they are
probably cosmological in origin. If they are at such large distances f
rom Earth then they are extremely luminous and compact. A consequence
of this is that two-photon pair production attenuation of the gamma-ra
y continuum cannot be avoided unless the source radiation is substanti
ally beamed. Most sources do not display gamma-ray turnovers although
a few GRB detected by GRO exhibit distinct spectral breaks in the MeV
range. A derivation of the relationship defining of the degree of beam
ing in burst sources with spectral breaks due to gamma-gamma attenuati
on, as a function of source spectral index and break energy, is presen
ted. It is found that sources at distances of similar to 1 Gpc must ty
pically be beamed with bulk Lorentz factors of around 10(3)-10(4), ind
icating powerful bulk acceleration in bursts, although these Lorentz f
actors are reduced markedly for steep source spectra. Since the source
spectra are not strongly Comptonized, such beaming will blueshift the
gamma-gamma attenuation breaks to energies much higher than 1 MeV; an
absolute lower bound to the source bulk Lorentz factor is determined
from this additional constraint. This blueshifting suggests that those
sources with MeV breaks may not be cosmological, or that their breaks
are produced by a mechanism that dominates gamma-gamma attenuation at
these energies.