<p>This paper investigates a recently introduced notion of strong variational sufficiency in optimization problems whose importance has been highly recognized in optimization theory, numerical methods, and applications. We address a general class of composite optimization problems and establish complete characterizations of strong variational sufficiency for their local minimizers in terms of a generalized version of the strong second-order sufficient condition (SSOSC) and the positive-definiteness of an appropriate generalized Hessian of the augmented Lagrangian calculated at the point in question. The generalized SSOSC is expressed via a novel second-order variational function, which reflects specific features of nonconvex composite models. The imposed assumptions describe the spectrum of composite optimization problems covered by our approach while being constructively implemented for nonpolyhedral problems that involve the nuclear norm function and the indicator function of the positive-semidefinite cone without any constraint qualifications.</p>