Should I wait for my insurance to contact the other party or just file a claim with theirs directly?

Hey everyone, looking for some advice on how to handle an accident that happened yesterday (April 22). This is my first time filing a claim and any input is appreciated.

I was rear-ended while turning into a parking garage (AZ). There’s visible damage, dents, scratches, and some parts of the rear bumper are dislocated.

Here is where I'm at:

I already filed a "notice only" claim with my own insurance. I told them I don’t want to use my collision coverage yet (to avoid the deductible), but I filed a claim with the hope that they would contact the other party’s insurance for me.

My questions:

1. Should I wait for my insurance company to make the first move, or should I just call the other guy’s insurance myself and start a claim with them?

2. Is it faster to go through the other party's insurance directly, or does that make things more complicated?

3. Since it was a clear-cut rear-end accident and the other guy was admitted he was in a rush, is there any downside to me reaching out to his provider today?