Yes, the person should be able to make a claim without first contacting your or having a police report.

While a police report or crash report is preferred by an insurance company however it is not always necessary and especially for times when the police are too busy to show up and ask that the drivers just exchange information.

If you were at fault and the other party has your insurance information he or she should be able to contact your insurance provider's claim department and file a claim for the damages they sustained due to you being at fault in the accident. If a file is placed against your liability cover then a claims adjuster from your insurer should contact your for a statement about the accident and let you know that a claim was submitted against your policy.