We study random vectors of the form (Tr(A⁽¹⁾V), …, Tr(A^(r)V)), where V is a uniformly distributed element of a matrix version of a classical compact symmetric space, and the A^(ν) are deterministic parameter matrices. We show that for increasing matrix sizes these random vectors converge to a joint Gaussian limit, and compute its covariances. This generalizes previous work of Diaconis et al. for Haar distributed matrices from the classical compact groups. The proof uses integration formulas, due to Collins and Śniady, for polynomial functions on the classical compact groups.