Abstract: In hidden sector models, dark matter does not directly couple to the particle content of the Standard Model, strongly suppressing rates at direct detection experiments, while still allowing for large signals from annihilation. In this paper, we conduct an extensive study of hidden sector dark matter, covering a wide range of dark matter spins, mediator spins, interaction diagrams, and annihilation final states, in each case determining whether the annihilations are s-wave (thus enabling efficient annihilation in the universe today). We then go on to consider a variety of portal interactions that allow the hidden sector annihilation products to decay into the Standard Model. We broadly classify constraints from relic density requirements and dwarf spheroidal galaxy observations. In the scenario that the hidden sector was in equilibrium with the Standard Model in the early universe, we place a lower bound on the portal coupling, as well as on the dark matter's elastic scattering cross section with nuclei. We apply our hidden sector results to the observed Galactic Center gamma-ray excess and the cosmic-ray antiproton excess. We find that both of these excesses can be simultaneously explained by a variety of hidden sector models, without any tension with constraints from observations of dwarf spheroidal galaxies.