Is it better to pay the damage privately instead of filing a claim after being involved in a car accident?

Here's the problem with that. Yes, you can prevent your insurance rate from going up (if, in fact, the rate would even be raised; it's not automatic after an accident.) But now you've entered into a verbal agreement with a total stranger. The other driver might demand more than you think is reasonable. You might not want to pay the estimated repair costs. The other driver may come after you later with further damages. He could claim medical costs down the road. People often end up in small claims court because they didn't want to file insurance claims. At the very least, write up a contract that stipulates exactly what you have paid and that it constitutes payment in full. But you may also want to check your drivers handbook -- in some states it's illegal not to report an accident and your insurance company could drop you if they find out. You should ALWAYS notify the police AND your isurer to avoid a surprise at policy renewal time! They have their ways of knowing about accidents regardless of how secret you keep it. It's better to notify your insurance company immediately or else this would be considered Non-disclosure of facts and hurt your history even MORE than an At Fault accident may.