Coming from someone who has been hit by other drivers 10 times in 10 years this is what I learned:
If you go through your insurance company you pay the deductible and then they go after the other insurance company and reimburse you when it's all over.
If you go through the other insurance you obviously don't pay the deductible.
Given the situation it's understandable that you don't have access to the other person's insurance right away. So if you need non emergent medical attention like massages or taking advantage of the PIP benefits, you can just use your insurance claim number. Otherwise you can wait it out till you get the other guy's insurance information.
I was hit by a drunk driver, he fled the scene and luckily I had enough sense of my surroundings to burn his license plate into my memory. They arrested him a week later but in the meantime I paid my deductible and started the medical/car repairs. When it was over they reimbursed me my deductible and the courts figured out the rest for the drunk. The police were able to give me the other driver's insurance information when he was arrested if I needed it, but I had already started the claim through mine.
Some insurance companies will waive the deductible if you can provide the police report to show it was indeed another driver's fault.