Does an AC wave running through a wire produce EM waves?
Yes, an AC current in a wire emits electromagnetic (EM) radiation.

How efficiently it emits depends on how long it is. If it's a tiny fraction of a wavelength long, then it's very inefficient at emitting propagating, far-field, EM radiation. Most of the energy stays as near-field, non-propagating magnetic fields.

The business of crafting a wire so it emits EM efficiently is called antenna design.