Who are responsible for conduction of electricity in an iconic compound (molten form), the ions or the free electrons?
The ions carry the charge.
There are no free electrons in ions, whereas in metals there are delocalized electons that are not associated with any centre.
As you know, in the solid state, the ions are not free to move, and are rigidly held in an electrostatic lattice. By the way, have you ever tried to melt sodium chloride? I hope your teacher has demonstrated the reaction.