No, we are not aware of any Missouri law or statute that states that auto accidents on private property are determined to be no fault. The instructions for the uniform crash report for MO give notes on how an officer should report an accident if it is on private property without an address so it goes to show that police will come and take a report on private property and help to determine fault if it is unclear.
Missouri does have laws and uses the rules of pure comparative fault. Pure comparative fault allows your damages to be reduced by the percentage you are at fault in a loss. Insurers are allowed to investigate an accident and make a decision as to the percentage of fault of all parties involved. They will make voluntary offers based on this opinion. If you disagree with their evaluation, you can consult an attorney or turn over the matter to your own insurance company to handle under your collision coverage.
The MO Department of Insurance gives information on parking lot accidents which are many times determined to be 50/50 comparative fault incidents. This may be what the person you hit is referring to as no fault.
Many parking lot accidents end up being half your fault and half the other person’s fault the Missouri DOI states. Parking lots are considered private property where the property owner, not the state, places the lines and road signs. Some of your general road laws do not apply, but common sense does.
Since the driveway would not have signs but common sense should be used and you would have the reasonable belief that a person backing up would take proper care and lookout by looking behind and not backing into your car she will likely be found at fault. Unless your car was parked illegally, then there may be a percentage of fault placed on you as well.
If the driver no longer wants to pay for the damages and you are unable to put the claim through her property damage liability insurance then you could make a claim on your own collision coverage or take the at fault driver to court in hopes to getting a judgment against her for the cost of repairs.
In an accident such as yours where the person clearly backed into your vehicle and took responsibility for the accident then normally she would be held at fault. Since she believes she cannot be held liable in this situation due to it being on private property you may want to contact the police or the Missouri Department of Insurance (Consumer Insurance Hotline 1-800-726-7390) to get information on what exact Missouri laws would cover this situation so that you could give the driver this information. Once she has this data hopefully she will take responsibility and pay for your damages.