We present the universal theory of arbitrary, localized impurities in a confining paramagnetic state of two-dimensional antiferromagnets with global SU(2) spin symmetry. The energy gap of the host antiferromagnet to spin-1 excitations, Δ, is assumed to be significantly smaller than a typical nearest neighbor exchange. In the absence of impurities, it was argued in earlier work [Chubukov et al., Phys. Rev. B 49, 11 919 (1994)] that the low-temperature quantum dynamics is universally and completely determined by the values of Δ and a spin-wave velocity c. Here we establish the remarkable fact that no additional parameters are necessary for an antiferromagnet with a dilute concentration of impurities, (Formula presented)—each impurity is completely characterized by a integer/half-odd-integer valued spin S which measures the net uncompensated Berry phase due to spin precession in its vicinity. We compute the impurity-induced damping of the spin-1 collective mode of the antiferromagnet: the damping occurs on an energy scale (Formula presented) and we predict a universal, asymmetric line shape for the collective mode peak. We argue that, under suitable conditions, our results apply unchanged (or in some cases, with minor modifications) to d-wave superconductors, and compare them to recent neutron-scattering experiments on (Formula presented) by Fong et al. [Phys. Rev. Lett. 82, 1939 (1999)]. We also describe the universal evolution of numerous measurable correlations as the host antiferromagnet undergoes a quantum phase transition to a Néel ordered state.