In many states you have the ability to enroll in the State Certified Police Academy as an "Open Enrollee" meaning you are attending on your own, not being "Sponsored" by a police department. There are many regional police academies that train new officers for a variety of difference police departments If they have space, they allow students to enroll on their own and pay their own way to attend. These students go through the same training as those attending that have been hired by their local police department.
The disadvantage to this is that you have to pay your own way, and you are not paid while you attend. In the past most new recruits when hired would be sent to the police academy and would be paid a salary while attending. The enrollment fees would be paid by their department. As departments are able to require more from job candidates, many departments will only interview or hire those that have gone through and have graduated from a state certified police academy. The advantage is that the department saves money by not having to pay the cadet a salary while in the academy, they do not have to pay the academy tuition, and the recruit is ready to start when hired.
This trend is increasing among police departments. To find out what academies are certified in your state, contact the POST (police officer standards and training) agency in your state. Only you can determine if it is worth going through an academy on your own.