A handful of non-blazar galaxies, including both Seyfert and starburst galaxies, have been observed to emit gamma-rays by the Fermi Gamma-ray Space Telescope. For one of these objects, NGC 1068, a starburst galaxy which is also the brightest and closest Seyfert 2, a galactic outflow driven by the active nucleus is a likely source of both the radio and gamma-ray spectra, and we compare this case to other possible explanations including starburst activity and blazar-like jet emission. Other galaxies with both starburst and Seyfert characteristics, including NGC 4945, NGC 253, Circinus, and NGC 3256, show varying degrees of evidence of starburst, AGN mini-jet and AGN-driven outflow. Given the existence of various possible emission mechanisms operating at high energies, we discuss prospects for observations of starburst/Seyfert galaxies with Fermi, HESS and the Cerenkov Telescope Array (CTA), as well as future neutrino telescopes such as KM3NeT.