Car crash: Insurance or take person at fault money?

IIRC, failing to report an accident resulting in severe damage (some minimum, usually $500) is illegal. I believe reporting the damage is mandatory in this situation.
Google "failure to report accident" and your state to determine the actual laws relevant to you.

I see that reasoning behind the law, but it also seems to me that if two adults want to deal with it on their own, perhaps they should be able to.

They are free to do that at the time of the accident. When the driver at fault damages their property and potentially causes injury to those in the other car and then flees in order to avoid civil/criminal liability, the police absolutely should get involved.

This is a very good point. I mean the guy committed a hit and run. He needs to face consequences other than just waving money around to make it go away.

There are also potential aggravating circumstances. What if, for example, he was driving drunk at the time of the accident and fled to avoid being punished for that? With the exception of some kind of medical emergency or a threat of imminent harm, there is no valid or lawful reason to not stop and handle it there.

Get the the proper channels involved. Now. There are a multitude of negative impacts here that you don't seem to be considering. As someone pointed out, failing to report the crime or details relating to the crime is a big no-no that could backfire on "Dave" and not just the person who hit his car. Also, $20k is a lot of money. You don't just magically get it. Now you're talking about tax implications and, if "Dave" is going to take the money as restitution while at the same time pretending a crime never happened you're getting damn close to tax fraud (I'm assuming if he didn't stop after totaling someone's car he likely isn't going to pay gift taxes on the money transfer either).
So yeah, call insurance and that police. Sooner rather than later.

California PI lawyer here. Just for the record, when most of the public thinks "oh, this is $20k in damage," what you actually have is a $50,000 lawsuit against an insurance company (through the at-fault driver).
Or "Oh, I just missed a few days of work and had a sore back for a few weeks" is actually a $10,000 lawsuit.
Failing to hire professionals from either your insurance (whom you've already paid for their professional help, by the way) or a lawyer is asking to get ripped off - or in legal trouble.

In Washington, the standard penalty for hit and run is license suspended for 1 year plus probation. So even if it's the first time the guy did it, he'll be in trouble.

If I were Dave, I would absolutely file a police report and sue Frank (and their insurance company) in a court of law. Dave's car might be totaled and $20k might cover that, but what about potential medical expenses from injuries resulting from the collision?
Dave needs to get a lawyer, retain ALL evidence showing that Frank is at fault, and not talk about this case on the internet. If I were Dave, I would also not tell Frank that he is doing so. Frank will find out when he's served.

Well deciding to use insurance is optional, and people settle things with cash all the time.
And the irony is, Dave is actually breaking the law (probably, but it depends on the state he lives in) by failing to report the accident (if your car has >$1,000 of damange you need to report the accident, not the other guy) to the authorities. Plus, depending on the state again, recording a phone call without the other party's consent is illegal (think Linda Tripp in the Monica Lewinski scandal) so he'd be breaking the law again :)

Is using insurance optional? I've been told that settling with cash and not reporting an accident is considered insurance fraud. But I'm not sure if that's illegal or just against the terms of my insurance policy.

I'm by no means an expert in these matters, but my common sense tells me that the not reporting bit would be illegal and that the insurance company would probably actually prefer if you paid in cash, since it would save them money.

Filing a claim (what I meant by "using insurance") is optional and not insurance fraud as far as I know, because remember the insurance company would prefer that you never file a claim. Worst case is they lose hundreds of thousands of dollars - and even in the best case scenario, when you're not at-fault at all, they're still out some money from having to pay an employee to deal with the other insurance company. Insurance fraud is more or less the opposite - it's when you claim something happened (or there was some extra loss) that was never there.

This is just from my experience, so do not quote me for this or assume this is proper advice, but my brother was in an accident for less than the deductible, and the insurance advisor just said to pay for it out of pocket since the costs would be higher with a claim. Also, this accident was reported to the police on scene.

Depends on the state you are living in, but California law is anything over a certain amount in damages must be reported to insurance.
Edit, over $750 in CA..
When you have a collision, report it to the DMV using the SR 1 form. You or your insurance agent, broker, or legal representative must complete the SR 1 report and send it to the DMV within 10 days if someone is injured (no matter how minor the injury) or killed or property damage is over $750. The SR 1 report is required in addition to any other report made to the police, CHP, or your insurance company.

The police report is the most important part. Where your friend gets the money is largely irrelevant as long as he gets enough to cover the cost of repairs and medical expenses.
That's a general statement though. I doubt Frank is a very trustworthy guy considering he bailed after the accident. Talk to the police and follow their advice.

Go through insurance, I got rear ended on my bike the driver pulled a runner. A witness gave me his plate, the police tracked him down and basically said to sort it out through insurance.
After contacting the driver he wanted to get the matter settled without the insurance companies getting involved, I was okay with this as the damage was fairly minor. I gave him the quote which was ~$350 for a new tyre and a number plate cover. He came back to me and offered $10 for the number plate cover claiming the tyre was not his fault then ceased to communicate.
When I went over to his house he wasn't home and his wife did not even know about the accident. As I was riding home he calls me up and asked me why I did I go to his house and that I could go and fuck myself, he was not paying. Luckily my uni has a free legal service and I now taking legal action against him.
But yea, go through insurance so Dave can get his car fixed and avoid all this crap. If it was a few hundred I might consider letting the guy pay, but $20K? no way.

First off, that really fucking sucks for Dave. I went out to my car one morning and the back door had been totally smashed in by another car, its a truly shitty and helpless feeling. BUT I never found the person at fault. $20k in cash is a nice chunk of change. Sounds to me like Frank knows he's at fault and reporting this could really fuck him over. It might be easier for Dave to just take the money. Like on the elementary school playground: no reason to go to the yard duty if you don't have to. If it was genuinely a mistake and Frank had an extreme moment of idiocy, then reporting this could fuck his life up. Felonies have a way of doing that to somebody.
That being said, a hit and run is REALLY shitty. If the damages are severe and this Frank character seems like a scumbag, it would probably be the right thing to report him. It's hard to say without more details but if Frank is a guy who made a massive mistake and is willing to try and handle this between men, I think $20k cash is a decent place to start.
Turning him in and suing him (like a lot of people seem to be in favor of here) could end up wreaking havoc on Frank's life more so than a totaled car. Just to repeat so I don't sound like an asshole: it is entirely possible (even likely) that Frank is a fucking dick, caused some serious damage and the world may be a better place with him in jail. Also if there was any damage to Dave himself or loved ones, or any person for that matter, calling the police is a really good option.
TL;DR Frank pulled a dickhead, spineless move and needs to pay for it. Particularly if there was damage to anybody who happened to be around. Slapping him with a felony and suing him could very possibly ruin his life. Without more details its hard to say if he deserves that. Also, 20 g's in your pocket is mighty tempting.

Why does everyone suggest suing? Just let the insurance companies handle it all. Depending on whether you're in a no-fault state or not, either his insurance company or your insurance company should take care of any lawsuits, right?

Hell, even in a fault-state, this is what uninsured motorist insurance is for. He'd have been better off not tracking down the perp. Uninsured motorist insurance (as far as I know) is pretty standard across all auto insurance types across all states, isn't it?
And of course if you're in a no-fault state, and don't provide your own collision coverage, then hell yes, take the money, because you probably can't sue, because it's no-fault, and you should have insured your own property.

No fault would also make your own insurance rates go up.

I'm not so sure. There are lots of conflicting studies and it's based on coverage. For strict PL/PD (i.e., no coverage for your own vehicle) rates are usually cheaper. If you're driving something worth less than $10,000 or so (and have money to replace it), straight PL/PD is a dirt cheap bargain, even in a no-fault state. It's because you're covering your own car, and you have no liability to cover the other car that you damage.
When I have full coverage (and I always use $10,000 as my own threshold) I'm perfectly content with my rates in Michigan. It's only about $1200 per year when the car is new, and goes down a bit per year until I drop collision finally. Of course there are lots and lots of variables in my rate, but it's dirt cheap for me.

I meant it would make your rates go up if you made a claim. I wasn't that clear there.

That's potentially true. I like my auto club because in this case it's not true. Unless I'm more than 50% at fault. Actually I think the language is really 50% negligent. I'm not sure if that's the law in my state, or just a benefit of my specific provider. Insurance is a confused mess in just my state, let alone across all 50 + D.C.