Stochastic partial differential equations are utilized to model a variety of physical phenomena such as heat distribution, fluid flow, wave movement or plate bending. The governing parameters in the underlying equations, however, may be unknown and statistical tools are required to identify them. In this paper, we aim to estimate elasticity and damping coefficients in general second-order stochastic Cauchy problems by means of multiple spatially localized measurements, which are part of a natural observation scheme. Joint asymptotic normality is established and the derived convergence rates coincide with the ones in the spectral approach. The arguments are based on ideas used in the nonparametric wave equation and convection-diffusion equations, but require major generalizations and functionalanalytical extensions in detail.