What insurance coverage is accepted at any given hospital is a policy decision. Having insurance is no guarantee that it is accepted at the facility you stumble into.
Nonwalk-in, emergency treatment at private facilities without insurance details being available at the time is limited to stopping the bleeding, making sure the patient is breathing. If you are conscious you will be asked for insurance details or a payment method before corrective treatment begins. If you are not conscious and there is no one to provide insurance details or to put down a deposit, your treatment might be limited to keeping you stable until such details are worked out.
Required life saving treatment is supposed to be provided everywhere and the payment details worked out later. Non life threatening injuries regardless of how painful or urgent may not be so quickly addressed. When in doubt a smaller facility will usually try to pass a patient off to a larger facility if the odds are good enough that the patient will arrive there not dead.
I don't know anything about insurance available to the common man here. If they can afford a gringo type policy, they are probably ok just about anywhere. If not, then it's possible that not every private clinic or hospital will accept and treat locals beyond what their policy covers or they can pay for.
I'm often on my own when out and about. One of the misgivings I have is being run down on my moto and delivered to CMC (about the only option here in Sosua) in a condition unable to communicate. I don't want people standing around waiting for me to wake up before getting down to business. I have a photocopy of my insurance in my wallet (which will be the first thing that goes missing), with my moto papers, inside my helmet and a laminated card in my pocket. I have also stopped into CMC and had them open a file for me and my wife. The only thing currently in that file is our insurance info and two emergency contact numbers - one for a person who speaks Spanish and the other my wife whose Spanish is limited to yelling at me to "stop drinking and get to work".