It is known that cosmic rays are efficiently produced at starburst galaxies (SBGs) with enhanced formation of massive stars and ensuing supernova explosions. Inelastic collisions of cosmic ray protons (CRPs) with background thermal protons are expected to generate gamma-rays and neutrinos, especially in the starburst nucleus (SBN). Considering uncertainties in nonlinear diffusive shock acceleration for the CRP production and possible scenarios of CRP transport out of the SBN, we adopt various models for the production and transport of CRPs and we estimate the gamma-ray and neutrino emissions from nearby SBGs, such as M82, NGC253, and Arp220. We identify the combinations of the production and transport models that can reproduce the gamma-ray fluxes observed with Fermi-LAT, Veritas, and HESS. We then estimate the neutrino fluxes and suggest that M82 and NGC253 might be detectable as point sources in the upcoming KM3NET and IceCube-Gen2 when the most optimistic models are employed.