Travelling in the opposite direction, does not mean kinetic energy is lost.
Whatever are their initial directions of motion, the final directions depends only upon their initial moments.
The loss in kinetic energy is not due to collision, but because of friction etc.
If it is said in the problem that the collision is elastic, we have to take that there is no loss in kinetic energies; if it is inelastic we say that there is loss of kinetic energy.
Some of the energies are lost to the surroundings. Some of the kinetic energies might have been transformed into some other form of energies such as sound, heat etc.
During collision the one having greater momentum looses momentum and the other gains momentum. The magnitude of gain by one = the magnitude of loss of the other.
Since momentum has direction involved in it, in a straight line, we take one direction as positive and the reverse as negative.
Since the one which has greeter momentum looses momentum, its magnitude may reduce to zero or even go to a negative value. In the latter case its direction is reversed.
Similar thing can be said of the other one which looses momentum.
Thus the final direction depends on how much it has lost or gained momentum.
Hence one cannot say that they will travel in the direction of the object having grater momentum.
One can say that the one having greater momentum will lose momentum and its final direction depends on how much it has lost its momentum.