When cops screw up they are put on paid leave. For the rest of us, paid leave is called vacation time.

No. If a cop makes a big mistake, they are generally fired. However, if they make minor violations of policy, they may keep their job but be reduced in rank. That reduction in rank could move them to a previous position. However, usually if this happens, you are reduced to working a desk job. The logic being that you don't deserve to be fired, but they don't want you out in public where your violations of policy can do harm. As for barbergc's comment, he is only partially true. A cop is a public employee who can only be fired if provided due process. However, a cop is only put on paid leave when there is an unsubstantiated accusation of wrong doing, or an administrative policy that requires investigation. The reality is that most cops will have many false accusations made against them in their career. If there is no evidence that wrongdoing occurred, it would not be right to suspend the cops pay until the accusation is corroborated. On the other hand, if there is corroborating evidence that misconduct occurred, the cop will be suspended without pay, and will get back pay if cleared of the accusation. As for admin reasons, a cop involved in a shooting is put on paid leave and investigated as a matter of course. These shootings are almost always legal, but are investigated nonetheless.

Demotions are generally the way to go, but it depends on how big the mistake was. My county is so militant that they fired the best cop they had because he told his superiors about another cop in possession of cocain the next morning instead of the night before when he discovered it.