We report numerical simulation results for free-vortex lifetimes in the critical region of classical two-dimensional easy-plane ferro- and antiferromagnets having three-component order parameters. The fluctuations in the vortex number density in a spin dynamics simulation were used to estimate the lifetimes. The observed lifetimes are of the same order of magnitude but smaller than the characteristic timescale above which a phenomenological ideal vortex-gas theory that has been used to account for the central peak in the dynamic structure factor is expected to be valid. For strong anisotropy, where the vortices are in-plane, the free vortex lifetimes for ferromagnets and antiferromagnets are the same, while for weak anisotropy, where the vortices have non-zero out-of-easy-plane components, the lifetimes in antiferromagnets are smaller than in ferromagnets. The dependence of the free-vortex and total vortex densities on the size dependent correlation length in the critical region is examined. We also determined the lifetimes of vortex-antivortex pairs for and well below . The observed timescales are very short, and the observed pair densities are extremely small. These results suggest that pair creation and annihilation are unlikely to play any role in the central peak in observed in computer simulations for the ferromagnetic model for .