The construction of local escape cones and their complementary cones, related to the locally nonrotating frames (LNRFs) orbiting superspinars with the external field described by the Kerr-de Sitter (KdS) naked singularity (NS) spacetimes, is described for all possible classes of the KdS NS spacetimes and all possible positions of the photon emitter. The notion of the local escape cones is then applied to construct the shadow of the KdS superspinars related to the distant static observers represented by the LNRFs located near the so-called static radius where the spacetime is close to the asymptotically flat region of the Kerr spacetimes. The shadow construction is focused on the KdS superspinars limited by the cosmological and astrophysical restrictions on the spacetime parameters, namely, the superspinar dimensionless spin and the dimensionless cosmological parameter relating the cosmological constant and the superspinar's mass.