No, the current stays inside the wires, but the charged particles inside the wire (the electrons) that carry the current can surge backwards and forwards around their rest position. (For an analogy, you may have sloshed water back and forth inside a bathtub as a child. The water doesn't leave the bathtub but it oscillates back and forth.)
When charged particles like electrons are made to oscillate back and forth inside a wire they cause power to be radiated away from the wire. (Note that what is radiated is power, not current. All the current stays inside the wire, all the time. If you did a charge balance you would find all charge accounted for and no electrons escaping.)