Sub-galactic astrophysical structure provides probes into the microscopic nature of the dark sector and its dark matter content. For example, dark matter with sizable self interactions and / or dissipation can leave distinctive signatures on the properties of satellite galaxies around Milky Way-like hosts. Recently, we placed new constraints on self-interacting dark matter models by analyzing Milky Way dwarf galaxies. The results push these models into a parameter space with a very specific prediction: self-interactions within satellite galaxies can be either very large (so large that new dynamical effects become important), or very small (so small that such models are usually thought of as collisionless), but not intermediate. If self-interactions are large, some dwarfs of the Milky Way must be undergoing a process of gravothermal collapse, and this has distinct observational predictions which can be searched for in current and upcoming data. In this talk I will discuss a path to fully probing the parameter space of self-interacting dark matter models by understanding these effects in realistic environments of Milky Way dwarf galaxies.