My friend is away from home right now; he lives in Ohio and has had a car accident in the state of New Jersey. What does he need to do locally before he can return home? He says a state trooper took his testimony; should he stick around and ensure a police report is filed or can he leave and handle anything else via mail?
He should call the courthouse for the relevant jurisdiction. If it was a state trooper that took down his info, then that means call the state courthouse. They can probably look up the incident based on his name, driver's license number, etc. Then ask them what will be required of him, and if he can return home.
Worst case scenario (if he wasn't already arrested for a grievous violation such as a DUI), is he'll be asked to appear in court later, if the police report says he is at-fault, and he is charged with a violation. He can probably waive his right to a court hearing, by mail (depending on what he is charged with), and pay whatever the default penalties are.
If he is not at-fault, then he probably doesn't need to do anything in New Jersey, from a legal standpoint. His insurance company will likely want to inspect the vehicle. If it's undrivable, that means in NJ, but he probably doesn't need to be around for that.
The only reason I can think of he may need to stick around (or return) to NJ if he is not at-fault is if the state wants him as a witness in prosecuting the other driver(s). And that will likely take several weeks or longer.
But there are really just way too many variables to provide any sort of definite answer, short of calling the appropriate courthouse and asking.
First, determine if emergency medical assistance is needed. Then, take pictures and get contact information from any bystander that witnessed the collision. Do not assume that the police/trooper will be thorough here -- they have many other accidents to work, crimes to solve etc. Second, tell the officer exactly what happened and make sure his/her version of events shows up in the police reports. Call your own insurance carrier to report the accident while on the scene to make sure you've done what your insurer may require. Typically, it takes 3-5 days for a police report to become available so waiting is usually not an option and many states now have there reports online. It should not matter that your friend travels out of state as long as this information is secure.
For more useful information, see McAleer Law's page "what to do after a car accident" -- it has a wealth of useful information.
The normal procedure is that trooper or a local police officer write up a report and mail it to their address. Problem is that they forget so what your friend needed to do was take down the information about the police officer. Department, Badge Number, and Name. Once he gets back to Ohio unless the report arrives in 2 weeks he will have to start calling up the department and asking for that report. The better solution would be to have his insurance company do that if he has comprehensive coverage on his insurance.
Thankfully he didn't total himself which has been known to happen on the back roads of the United States. :)