In this talk, we address the localization of general nonlocal functionals of double-integral type with fractional dependence on the state variable, inspired by peridynamics. Localization is carried out as the interaction horizon among particles tends to zero. As a main result, we obtain an explicit formulation of the local $\Gamma$-limit, also covering the vectorial case. Applications of this result to nonlinear elasticity and the p-Laplacian eigenvalue problem will be discussed.