You’re not required to. You do not have to speak with the other driver’s insurance company, and you certainly don’t need to give a recorded statement. There is no law that says you must speak with an adjuster of the other driver’s insurance company.
As far as whether you should speak with your insurance company after a bicycle accident, the answer is less absolute. Many insurance policies have clauses that say if you don’t notify your own insurance company relatively quickly after an accident, then your insurance company won’t have to compensate you. If you are uncertain whether you’re going to hire a lawyer, you may want to consider calling your insurance company and notifying them of the accident but be careful when you do. If you do decide to hire lawyer early on, then let your lawyer handle calling your insurance company.
We’ve also addressed this question in our page on Car Accident FAQs.