The insurance company adjuster is calling you, and they want to "talk." Do you answer?
Ok, you can answer the call. But there is no need to tell the insurance company anything. The insurance company is calling you for one thing. To figure out how they are going to minimize any payments to you.
That goes for the other driver's insurance company, as well as your own insurance company. Yes, your own insurance company knows there is often a chance they will have to pay you for your injuries as well. If you carry Uninsured Motorist Coverage, your company may be on the hook to compensate you for your medical bills and pain and suffering.
So when your own adjuster calls you, go ahead and answer. But be very careful what you say. Eventually you may be required to give a statement under the terms of your policy. But that statement can be given later, and there is no reason to tell your insurance company everything right away. It is okay to take your time and think things through.
As for the adjuster for the other driver's insurance company, there is no rule or requirement that you need to speak with them. In fact, you should not speak to the adjuster at all. If you are in a minor accident and don't think you need an attorney, then go ahead and try to settle your case by yourself, but there still is no need to speak to the adjuster. You can tell them, you will be corresponding strictly by letter. Just send in your demand package when you have completed your medical treatment.
It does not pay to speak to the other insurance company.