If someone hit you on private property and does not want to go through insurance should you report it anyway?

If someone hit you, it is their liability not yours. As long as there were no injuries and you are comfortable having the repairs made by a reputable body shop and not having the company involved, it's fine. However, if there is subsequent damage to your vehicle, or you are injured and you don't report it to their insurance company, you could have problems collecting reimbursement after the fact. Especially if they had you sign a settlement agreement. Just be careful!