The fundamental groups of hyperbolic surfaces and 3-manifolds, referred to as surface groups and 3-manifold groups, respectively, have various algebraic finiteness properties. Two of these properties, residual finiteness and subgroup separability, have played an important role in the recent resolution of some outstanding conjectures in 3-manifold theory. We will explain how effective proofs of these properties can help us "quantify" separability and discuss the topological implications of such quantifications. We initially focus on the hyperbolic surface case and then discuss joint work with K. Bou-Rabee and M.F. Hagen extending effective separability results to hyperbolic 3-manifold groups using right-angled Artin groups.