The spin dynamics of an arbitrary localized impurity in an insulating two-dimensional antiferromagnet, across the host transition from a paramagnet with a spin gap to a Neel state, is described. The impurity spin susceptibility has a Curie-like divergence at the quantum-critical coupling, but with a universal effective spin that is neither an integer nor a half-odd integer. In the Neel state, the transverse impurity susceptibility is a universal number divided by the host spin stiffness (which determines the energy cost to slow twists in the orientation of the Neel order). These and numerous other results for the thermodynamics, Knight shift, and magnon damping have important applications in experiments on layered transition metal oxides.