If the accident was minor, you and the other drivers may decide to handle the damages yourselves without the involvement of an insurance company. But this isn't always the best idea for several reasons.
While the other driver may agree to pay for the damage to your car on the day of the accident, he may see the repair bills and decide it's too high. At this point, time has passed and your insurance company will have more difficulty piecing together the evidence if you file a claim.
Also, keep in mind that you have no way of knowing whether another driver will change his mind and report the accident to his insurance company. He may even claim injuries - such as whiplash - that weren't apparent at the scene of the accident and show up the next day. This means that your insurance company may end up paying him a hefty settlement or worse yet, you could be dragged into a lawsuit.
Make sure that your company has your version of what happened and check your policy -- if the damages paid out by your insurance company are below a certain amount, the accident may not be considered chargeable. And you will avoid the penalty of a premium hike.