Is an at-fault drivers insurance company legally required to contact me?

If you have filed a claim or a loss report with the Insurer of the At Fault Driver, then "yes" they are required to respond to your claim. The time limits imposed on their response will vary depending on your state Insurance laws. Each state regulates the insurance industry as operates within it so the rules will vary from state to state. An Insurer however is generally not required to contact you if no claim or loss notice has been presented to them.