It really depends on the department. Some smaller PD's only have one or two handlers and no "trainers," and the handlers go and pick the dog from a vendor. In our Department, myself and another trainer select the potential K-9's. We do bring the new handlers with us and they watch the evaluation testing and selection process. We will narrow it down to the most desirable dogs and rank them. We will then allow the new handler some input and discuss his / her options. I do take into consideration matching the right dog with the right handler, even family considerations. But, at the end of the day we are not looking for "Lassie." I want strong dogs and will make the final decision. New handlers that have a preference for breed, color or size do not get to choose.
A few years ago, I had a new handler that was a GSD guy and wanted a GSD in the worst way for his K-9 partner. I told him before we left on our trip to test dogs, that he probably wouldn't get a GSD, but he would get a good "working dog." He got a Dutch Shepherd and was thrilled.
I know other agencies where the trainers select the dogs and pair them with the handlers; the handlers have no say in the selection process.