As the conduction electrons are free to move randomly, i.e they are not restricted to a particular atom, then why don't these atoms get positive charge due to loss of these conduction electrons?
You can think of the electron motion in the conductor as being something like this:
So no atom is left without an electron. In effect the conduction electrons hop from atom to atom.
I must emphasise that this isn't really what happens because the conduction electrons are delocalised so they are spread out over distances much larger than the atomic spacing. If you're interested in pursuing this the electron states are approximately described by Block waves. However as the simplified diagram shows, every atom is still on average associated with one of the conduction electrons so on average remains neutral.
To expand a bit on the previous answer, the behaviour of electrons in a metal is approximated by the Free Electron Model. All metals, that are good conductors, have only one or two valence electrons (i.e. in the outer electron shell). In this model, we imagine that the valence electrons leave the atom and the remaining ions form a regular lattice of some arrangement. The free electrons then form a sea or a gas that is free to move through the crystal. On average, the whole structure remains electrically neutral.
This model is very accurate in representing many of the bulk properties of metals - like density and electrical and thermal conductivity.