You say it will transfer the kinetic energy, but that is only partly correct. In the situation that you describe, the energy that the spinning, rolling, ball transferred was only the forward motion, plus a fraction of the angular momentum. We know that some of the angular momentum still remained in the ball, because the ball was still spinning forward. By engaging with the surface of the pool table, the ball then converted more of the angular momentum into forward motion, possibly enough to enable it to catch up with the ball that it had originally accelerated forward. 

Let us start by hitting the cue ball dead centre, no top spin, no bottom spin and no side spin. The ball starts by sliding along the surface, gradually it will start to rotate because friction slows the bottom of the ball but not the top. The harder the ball is hit, the further it will go before the rotation becomes obvious.

If the ball is hit hard and it is only a short distance before it hits another ball, and lets make that a head on hit, there will be very little rotation of the cue ball and it will look as though it stops dead and the other ball goes of at the same speed that the cue ball was moving. In this instance we can say that all of the energy from the cue was transfered to the cue ball as kinetic energy and intern the cue ball tranfered it as kinetic energy.

If the cue ball is not hit so hard or it travels a bit further before striking the other ball, the cue ball will start to rotate or roll. In this case the initial energy from the cue will be lost as friction, stored in the rotation of the ball and stored as kinetic energy. When the cue ball strikes the target, the kinetic energy is transfered to the target ball but the energy stored in rotation will keep the cue ball rolling until friction converts it to heat.

Now, if you apply spin to the cue ball by hitting it off centre you will change the magnitude and maybe direction of rotation of the ball when it hits the target ball. The kinetic energy will be transfered to the target and the energy stored in the rotation or spin of the cue ball will determine movement of the cue ball after it strikes the target.

Some of the energy stored as rotation in the cue ball will be transfered as rotation to the target ball. There is very little friction between the balls and so this transfere of energy is very small and, unless you hit things very hard, not noticable.

I hope that wihout diagrams and yards of calculations this describes roughly what is going on.

It is quite common to smack the cueball low down or off centre so it gets forward motion but back and/or sideways spin.

If it is still in this condition when it hits the target ball (i.e. the table friction has not yet dissipated the back-spin) then the hit ball goes forward (obliquely if it was hit off-centre) but an expert player can make the cue ball go almost anywhere he wants independently. Most competent players can pot a ball. 90% of the expertise lies in putting the cue ball where another good position can be played. Using back spin to get the cue to a cushion where the side spin comes into play is particularly approved.

Incidentally, complete transfer of the kinetic energy requires a direct hit, but also that the balls weigh the same. Pool cue balls are made slightly smaller and lighter (more so in the USA) so the table mechanism can return a potted cue ball but retain the target balls. This changes the behaviour of the balls quite markedly compared to snooker balls.