I've done this experiment myself and seen it front of my eyes. Set up a basic circuit with a battery, an LED, and connect the circuit with a salt-water solution and copper wire. The LED lights up.
If a potential difference is applied across two ends of the wire with the battery, positive charges move towards the end with lower potential and negative charges towards the higher potential.
Why does this conduction keep continuing? Shouldn't there eventually be a pile up of Na+ and Cl- on opposite ends to cut off the circuit? This is with a typical 9V battery. Is there a teensy bit of electrolysis going on in the wire that continuously supplies electrons to the circuit?
EDIT: In doing my own reading the next day, I seem to have forgotten the very important fact that water itself is electrolyzing at the surface of the electrodes. The salt water only helps to prevent the resulting electrolysis from quickly shielding the electrodes. The reason I was confused was because I was very skeptical of NaCl being electrolyzed and turns out this is exactly not the case