Left-handed and right-handed relief specialists are intended to mimic how managers in the MLB use them today. Specialists are brought in in the 6th inning or later when they can take advantage of the lefty/lefty or right/righty pitcher advantage.
The first scenario when they will be used is if it's a late and close situation where the out is paramount. If getting one (or a series of like-handed hitters) would benefit from a specialist, he'll enter the game.
The second scenario is if it's a close game in the late innings (not necessarily a dire situation) but there is a string of same-handed hitters coming to the plate.
Non-fatigued specialists will remain in the game until an opposite handed hitter comes to the plate. If a switch-hitter comes to the plate but the next hitter is like-handed, he'll face the switch hitter so he can stay in and face the next hitter unless it's a dire situation.
Specialists can face hitters of the opposite site when the manager doesn't have any other alternatives (such as it's the 7th inning but no one else is available until the 9th). This can result in a specialist facing more opposite-handed hitters than like-handed hitters over the course of the season. It's all about the manager having the necessary options available in the pen.