I recently completed the main storyline of LA Noire, and now I am still playing it to roam the streets and take on the random street crimes that happen.
There are hidden special cars in garages scattered throughout the game, as well the ability to pretty much walk up to any car and commandeer it. When you drive a new car, you are given a little notice that tells you that you've "unlocked" a new car.
However, since only police cars have radios, you won't come across any street crimes. So you can only drive around in them, but LA Noire isn't really set up to have any challenges that are based on driving. As far as I can tell, you literally just drive around a bit in them.
Since I finished the game without ever even knowing hidden cars existed, and only once commandeered a car (and I don't think it was strictly necessary in that situation), I'm not sure I understand why I would want to ever drive any car other than a police car.
Is there anything a player can do with a non-police car? Are they just for "collecting" and that's it?