Your best bet is to contact an attorney. You may even consider contacting the police department, but I would recommend contacting a legal representative first since an attorney should be able to assist you in getting this other party to cooperate.
It's never a bad idea to carry as much liability coverage as you can afford to protect yourself against lawsuits. That does not necessarily mean that you need a $500,000 liability limit, but you very well might.
Technically, the car needs to be insured before you get behind the wheel, even if the vehicle isn't registered yet in your name. The main issue in your situation is, if you get pulled over on your way home, you could be given a ticket for driving without insurance.