  The principal paradigm for the prompt emission of gamma-ray bursts invokes
diffusive acceleration at shocks internal to the dynamic ultrarelativistic
outflow. This paper explores expectations for burst emission spectra arising
from shock acceleration theory in the limit of particles cooling much slower
than their acceleration. Parametric fits to burst spectra obtained by the
Compton Gamma-Ray Observatory (CGRO) are explored for the cases of the
synchrotron, inverse Compton and synchrotron self-Compton radiation mechanisms,
using a linear combination of thermal and non-thermal electron populations.
These fits demand that the preponderance of electrons that are responsible for
the prompt emission reside in an intrinsically non-thermal population, strongly
contrasting particle distributions obtained from acceleration simulations. This
implies a potential conflict for acceleration scenarios where the non-thermal
electrons are drawn directly from a thermal gas. It is also found that the CGRO
data preclude effective spectroscopic discrimination between the synchrotron
and inverse Compton mechanisms. This situation may be resolved with future
missions probing gamma-ray bursts, namely Swift and GLAST. However, the
synchrotron self-Compton (SSC) spectrum is characteristically too broad near
the $\nu F_{\nu}$ peak to viably account for bursts such as GRB 910601, GRB
910814 and GRB 990123. It is concluded that the SSC mechanism may be generally
incompatible with differential burst spectra steeper than around $E^{-2.5}$
above the peak, unless the synchrotron component is strongly self-absorbed.
