Is it easier to go through your own insurance company or contact the at-fault drivers insurance directly?

That depends on your definition of "easier" considering the time, money, and hassle in fixing a vehicle. As well, it's always best NOT to have a claim showing on your record - your fault or not! Having said that, on the other hand, NEVER lie to your ins. company about an accident, even if you get it fixed yourself. That is grounds to drop you. I'm a broker trust me. It is wiser to contact your insurer first. It is considered material misrepresentation to avoid contacting your insurer after an accident. They will represent your claim, and if the other party is indeed At Fault, your insurer will subrogate to get funds back that are paid on your behalf.