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, NGC 253, and Arp 220. We find that with the current gamma-ray 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 a single power-law (PL) momentum distribution for SNR-produced CRPs and a diffusion model in which the CRP diffusion is mediated by the strong Kolmogorov-type turbulence of delta B/B similar to 1 , and (2) the combination of a double-PL model for SNR-produced CRPs and a diffusion model in which the scattering of CRPs is controlled mostly by self-excited waves rather than the preexisting turbulence. The contribution of SW-produced CRPs could be substantial in Arp 220, where the star formation rate is higher and the slope of the initial mass function would be flatter. We suggest that M82 and NGC 253 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.