In most cases, it's better to file the claim to fix your car with the at-fault party's insurance carrier as opposed to your own carrier, and here's why. You get a rental vehicle for the period of time that your car is being repaired. Secondly, and perhaps more importantly, you also have a claim against the at-fault party for what's called diminished value. A vehicle, if it's been involved in a motor vehicle accident - even if it's fixed perfectly - is not worth as much as a vehicle that's not been involved in an accident. A diminished value claim allows you to recover from the at-fault party's insurance company the difference between a car that hasn't been hit and a car that has been hit.
However, a claim against the at-fault party's insurance carrier usually takes longer. That's particularly true if they're disputing liability, so you may want to consider filing the claim on your own collision coverage if you have it. The benefits of that are that it's faster, usually less hassle, and you get it done without having to mess with it. Here's the downside though; if you file the claim under your own insurance, you have to deal with the deductible. Your deductible may be two hundred and fifty dollars, it may be five hundred dollars, maybe a thousand dollars, so you have to pay that up front. You may not get a rental vehicle unless you've purchased that coverage separately and there's no claim for diminished value.
So; do you file the claim to fix your car under their insurance, or under your insurance? Well, it depends on the facts of your particular case. We can help guide you through the best choice for you.