it appears your are in NJ? I have been an adjuster/appraiser for many years and can give you a little advice. You pay for insurance; shit happens, so use it. Why deal with the headache of fighting somoene else; if, and only if, your insurance can find the other party at fault they can recoup your deductible.
To be honest calling the Police does not matter as they were not there at the time of loss, and the report only is a tool for adjusters in determining liability. If this was on private property, then it really doesn't matter as vehicle and traffic laws don't specifically apply the same. You both have damages on the front of your cars? Sounds like two parties are at fault and you may share some of the liability. Sucks but it is what it is. It's hard to disprove fault when moving forward and striking something with the front of your car unless it was some unique circumstance.
PM me if you need any help