Don't let someone who hit you on your bike convince you that you don't need to call the police. They might want to exchange information and tell you things like, they'll take care of it or their insurance company will pay for all the damage. There are so many cases like this that once information was exchanged the driver just drives off and then later changes their mind or never contacts the cyclist back. They will then deny to their insurance company that they weren't there or it wasn't their car that was involved.
The bottom line is after a cycling accident you always want to call the police and you always want to remain at the scene until the police come and investigate. This is the best way to confirm what you said happened and the police will then be able to confirm the driver and their car was actually involved.