Nanocontact spin-torque oscillators are devices in which the generation of propagating spin waves can be sustained by spin transfer torque. In the present paper, we perform combined electrical and optical measurements in a single experimental setup to systematically investigate the excitation of spin waves by a nanocontact spin-torque oscillator and their propagation in a Ni80Fe20 extended layer. By using microfocused Brillouin light scattering we observe an anisotropic emission of spin waves, due to the broken symmetry imposed by the inhomogeneous Oersted field generated by the injected current. In particular, spin waves propagate on the side of the nanocontact where the Oersted field and the in-plane component of the applied magnetic field are antiparallel, while propagation is inhibited on the opposite side. Moreover, propagating spin waves are efficiently excited only in a limited frequency range corresponding to wavevectors inversely proportional to the size of the nanocontact. This frequency range obeys the dispersion relation for exchange-dominated spin waves in the far field, as confirmed by micromagnetic simulations of similar devices. The present results have direct consequences for spin wave based applications, such as synchronization, computation, and magnonics.