  Highly relativistic jets are a key element of current gamma-ray burst models,
where the jet kinetic energy is converted to radiation energy at optically thin
shocks. High-energy neutrinos are also expected, from interactions of protons
accelerated in the same shocks. Here we revisit the early evolution of a
relativistic jet, while the jet is still inside the star, and investigate its
neutrino emission. In particular we study propagation of mildly relativistic
and ultrarelativistic jets through a type Ib progenitor, and follow reverse
shocks as the jets cross the star. We show that protons can be accelerated to
10^4-10^5 GeV at reverse shocks, and efficiently produce mesons. The mesons
experience significant cooling, suppressing subsequent neutrino emission. We
show, however, that the neutrino yield from the reverse shock is still
reasonably large, especially for low-luminosity and long-duration jets, where
meson cooling is less severe. We discuss implications of our results in the
context of neutrinos from choked jets, which are completely shock heated and do
not break out of the star. From a choked jet with isotropic equivalent energy
of 10^{53} erg at 10 Mpc, we expect ~20 neutrino events at IceCube.
