I lost my US passport twice outside the US (marina near CPH airport and in Tokyo). I swiftly got them back from the police at the airport and the closest koban, respectively.
Generally speaking, I think it's a good idea to give it to the police near where you found it. If the person retraces his steps then it is the logical place to inquire. By the time it finds its way through the bureaucracy (via the mail system and Dept of State) the person will probably have obtained another one.
In Japan, I noticed the endless forms the poor fellow who found the passport had to fill out at the koban and there was a lot of paperwork and interrogation for me to get it back. When going through the forms they had a special template that they placed over the paperwork of the person who found it, in order to protect his identity. Unfortunately, the cop had spent a year in high school in the states, so could speak fluent English. It's a lot better when they can't speak English like when I got pulled over once.........