  We study high-energy neutrino production in collimated jets inside
progenitors of gamma-ray bursts (GRBs) and supernovae, considering both
collimation and internal shocks. We obtain simple, useful constraints, using
the often overlooked point that shock acceleration of particles is ineffective
at radiation-mediated shocks. Classical GRBs may be too powerful to produce
high-energy neutrinos inside stars, which is consistent with IceCube
nondetections. We find that ultralong GRBs avoid such constraints and detecting
the TeV signal will support giant progenitors. Predictions for low-power GRB
classes including low-luminosity GRBs can be consistent with the astrophysical
neutrino background IceCube may detect, with a spectral steepening around PeV.
The models can be tested with future GRB monitors.
