Answer: You do not have to use your own insurance. However, your insurance owes you a higher duty as their customer than the opposing insurance. As a result, you will typically see a faster response from your insurance as well as higher valuation of your vehicle if it is totaled.
Another reason to use your own insurance is if the opposing insurance is trying to place any fault on you. It is a common tactic that insurance companies will try to remove the blame from themselves. While we may ultimately sue over a liability dispute, if you do not use your own insurance you may be waiting months or even years for your vehicle to be taken care of. Ultimately, it will be less hassle for you to go through your own insurance, and they will seek reimbursement from the other party at a later date.