  Starburst galaxies (SBGs) with copious massive stars and supernova (SN)
explosions are the sites of active cosmic-ray production. Based on the
predictions of nonlinear diffusive shock acceleration theory, we model the
cosmic-ray proton (CRP) production by both pre-SN stellar winds (SWs) and
supernova remnants (SNRs) from core-collapse SNe inside the starburst nucleus.
Adopting different models for the transport of CRPs, we estimate the
$\gamma$-ray and neutrino emissions due to $pp$ collisions from nearby SBGs
such as M82, NGC253, and Arp220. We find that with the current $\gamma$-rays
observations by Fermi-LAT, Veritas, and H.E.S.S., it would be difficult to
constrain CRP production and transport models. Yet, the observations are better
reproduced with (1) the combination of the single power-law (PL) momentum
distribution for SNR-produced CRPs and the diffusion model in which the CRP
diffusion is mediated by the strong Kolmogorov-type turbulence of $\delta
B/B\sim1$, and (2) the combination of the double PL model for SNR-produced CRPs
and the diffusion model in which the scattering of CRPs is controlled mostly by
self-excited waves rather than the pre-existing turbulence. The contribution of
SW-produced CRPs could be substantial in Arp220, where the star formation rate
is higher and the slope of the initial mass function would be flatter. We
suggest that M82 and NGC253 might be detectable as point sources of high-energy
neutrinos in the upcoming KM3NET and IceCube-Gen2, when optimistic models are
applied. Future observations of neutrinos as well as $\gamma$-rays would
provide constraints for the production and diffusion of CRPs in SBGs.
