What happens if you run into the back of another vehicle and dont have insurance and the other driver says he will have it fixed privately and then changes his mind and goes through his insurance comp?

Well basically what could possibly happen is when his insurance kicks in and pays for his damages (assuming he has full coverage/uninsured motorist), is that you may find yourself facing a lawsuit from his insurance company demanding payment from you for damages. He also has the right to sue you privately if he has enough proof. In the end it's up to him how he gets his car fixed since you are 100% liable.