What type of insurance information do I need to submit?
Is there any consequence if I buy my car from a private owner's sale?
If that so, both you and owner need DMV to calculate taxes, in order to issue a payment receipt. This can be done through the filling of form DTF-802 for Tax and Finance by both parts.
You will need both your NY State Insurance ID cards and the electronic notice that your insurance company will provide to you. Both are required.