Dynamics of rotating objects is not always intuitive, therefore it is a good idea to do some thought experiments to help understanding what is happening.
How is spin transferred between Snooker balls?
At first sight this seems relatively simple, imagine that the cue ball (on the right) is moving with simple rolling motion, its angular momentum is therefore related to its linear motion.
It then hits the stationary object ball and an impulse is transferred between them.
After the collision the cue ball is stationary and the object ball moves to the left with rolling motion.
Since the object ball is moving in the same way as the object ball was then both linear and angular momentum are conserved then there would appear to be no problem.
However I think its worth investigating this more carefully. How is the angular momentum transferred between the balls? To transfer angular momentum in a collision we need two impulses which are not in line. However the tenancy would be to put backspin rather than forward spin on the object ball, for instance, if the cue ball is spinning in an anti-clockwise direction and is in contact with the object ball then it would tend to turn the object ball in a clockwise direction.
So why doesn't this happen? why does the object ball spin in the same direction even though the one of the faces of the balls will be going up and the other will be going down?
For further discussion of this question see this page.