If I have been injured in a car crash, should I give a statement to the other driver's insurance company?

This depends on whether you intend to hire a lawyer or attempt to handle your own case.

If you plan on handling your own case and want to get it settled without filing a lawsuit. the insurance company may refuse to settle with you without a recorded statement. They know that if you do not have a lawyer you are less likely to file a lawsuit, so they will generally put aditional requirements on you to settle, and they typically offer you less money.

If you intend to hire a lawyer, then it is best to wait until you are represented, to give a recorded statement. They are purposefully trying to reach you before you retain an attorney so they can get you to say something in a recording that will hurt your case. They will also try to get personal, private information from you that they are not entitled to. You should assume that anything you say can and will be used against you. Generally, insurance adjusters handle thousands upon thousands of these cases, so they know what they want you to say to help their defense of your claim. They are not trying to help you. They try to get these recorded statements early on, before you realize the full extent of your injuries. They do this  so they can use it against you when you say"my neck starting hurting a week after the crash," and they can pull out the recording and say you never mentioned  a neck injury in your recorded statement. They know injuries can take days, weeks, or even months to make themselves known to the injured person and they want to take advantage of that fact. They may even offer you money to settle immediately, because they do not want you to realize the full value of your claim, Thus it is better to decline a recorded statement and have a lawyer present with you when you do give one, for liability purposes only. so the injury can be pointed out on the recorded statement.