Ack! I got side-tracked. Are you talking about accidents on the streets or in the dunes? Either way the answers are pretty similar. By the way, I'm not dispensing advice, only expressing my opinions 8)
In my opinion, if
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No one is hurt and no one it to blame.. min property damage.
ALWAYS exchange information, no matter what. Keep a disposable camera in your glovebox for this kind of thing and take a picture, even if you can't see the damage. The reason is, plenty of times we've had clients get sued after shaking hands following a minor fender-bender and the crooks try to sue for damages that were never there.
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No one is hurt and both sides claim the other is responsible. Substantial property damage.
NEVER admit responsibility -- "geez, I'm so sorry, I didn't see you there" can translate into big money

Follow the same rule as above, making sure you get full and complete information. Since damage is substantial, call the authorities so they will have to make a record. Either way, notify your insurance company immediately just in case he does the same and his insurance company comes after you.
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no one is hurt and there is damage. I admit fault.. Am I responsible for damages?
Admit fault?? Ack! *faint* hehehe
Even if you are obviously at fault, don't say a word. But just in case you blurt it out, you're not automatically screwed... it will be up to a judge to decide if you're really at fault. If the authorities come out they may issue a citation, which could pretty much decide the issue, but there are ALWAYS mitigating circumstances (I'm sorry I hit him but he was wearing black, there was no moon, it was the dead of night and he threw himself i my path).
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Obviously if people are hurt the locals need to be called in. I assume they will assess a fault if there is one. I also assume if there is alcohol involved, that party will be at fault.?
Actually, it's not their job to assess fault, above and beyond citations for broken laws. They only get "just the facts ma'am" and if and when it goes to court the judge/jury decides fault. Remember, mitigating circumstances. Things are not always as they seem, so don't rely on the police to figure it out for you. That disposable camera can be what turns the case. Will the drunken bum always be at fault? Not if he has a good lawyer.
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What if you are hit causing damage but no injuries and the other person runs? what do you do? what can you do?
A runner probably has no license and no insurance. Your insurance may still cover the damages, depending on your coverage. Or it could be a scared kid who's afraid of telling his folks... hope they have coverage. Do the obvious, that is, take down description, make, model, dents, scratches (officer, it was a blue car with a primer side panel). Always look around you for witnesses. The courrier on the bike sitting at the light. The barber standing in the doorway. Maybe they saw something you missed. CALL THE POLICE. Even if there is no damage, hit and run is a crime. They may not come out "sorry buddy, we can't help" but they will have to make a record of the call. If they refuse to come out, go to the nearest police station and make a report.
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What if there are Injuries and they run?
Dial 911 and then start doing the other things.
Along with the disposable camera (a worthy $5.00 investment), keep paper and a pencil in your glove box so you can start recalling details right away. Things get fuzzy when the police arrive.
Try to pay attention. Accidents rattle your brain and your nerves. Compare names on insurance cards versus drivers' licenses and make sure the dates are correct. If you are shown an expired insurance card or driver's license, you should call the authorities because it could this guy's just not legit and you'll be screwed if you let him get away. If you call the cops, just explain that you've been involved in a minor accident with a guy who has no current proof of insurance and an expired license.
If the other driver involved doesn't cooperate, be nice and polite and call the authorities. If you're embarrassed about it, tell him you're on the job and your employer requires it.
Vicki