The properties of static, spherically symmetric configurations are considered in the framework of two models of nonlocally corrected gravity, suggested in S. Deser and R. Woodard., Phys. Rev. Lett. 663, 111301 (2007), and S. Capozziello et al., Phys. Lett. B 671, 193 (2009). For the first case, where the Lagrangian of nonlocal origin represents a scalar-tensor theory with two massless scalars, an explicit condition is found under which both scalars are canonical (non-phantom). If this condition does not hold, one of the fields exhibits a phantom behavior. Scalar-vacuum configurations then behave in a manner known for scalar-tensor theories. In the second case, the Lagrangian of nonlocal origin exhibits a scalar field interacting with the Gauss-Bonnet (GB) invariant and contains an arbitrary scalar field potential. It is found that the GB term, in general, leads to violation of the well-known no-go theorems valid for minimally coupled scalar fields in general relativity. It is shown, however, that some configurations of interest are still forbidden, whatever be the scalar field potential and the GB-scalar coupling function, namely, "force-free" wormholes (such that g_{tt}= const) and black holes with higher-order horizons.

11 pages, no figures. References and comments added, a final form to appear in PRD