We apply the principle of conservation of linear momentum. We cannot apply the principle of conservation of mechanical energy as, there is some loss of energy in deformation or sound energy.
let the two bodies get stuck to each other and travel with a speed v. They travel together with the same speed.
m1 v1 + m2 v2 = (m1 + m2) v
v = (m1 v1 + m2 v2) / (m1 + m2)
Suppose m1 = m2, then
v = (v1 + v2)/2
so their final velocity can be equal to initial velocity only if v1 = v2. Then there is effectively no collision.
Thus, after collision, the initial velocities cannot be equal to final velocities. But an exchange of velocities can happen. But then the bodies do not stick to each other.