I actually don't know how our school district identifies gifted students. I don't know what they would need to see to initiate their own testing. After all we have been through, I'm still not sure anyone there has a clue.
They do a kindergarten screening, MAP testing, STAR testing, DIBELS testing, State assessments, etc., but either none of that counts for anything other than school having a score recorded in their chart or they don't understand if a student is years ahead on every assessment they take maybe the school needs to look into that.
We had both of our daughters privately tested (for different reasons) then presented those results to the school. Luckily, they accepted our testing! Even with the testing and the recommendations stating they both needed additional services, the school still had to show there was a "need" in order to give us a GIEP.
Both girls topped out the kindergarten screening, have always received top scores in school, and have always been well above level on MAP and STAR testing. We repeatedly heard how advanced they were at parent/teacher conferences, but no one ever mentioned anything about the possibility of them being more than just advanced. DD10 showed definite, classic signs of academic boredom in 1st grade that led to school saying she was ADD/ADHD or ODD, which led to us getting her tested to rule out those things. That was how we found out she was gifted. If we hadn't done the testing I know DD10 would never have been identified. DD8 might not have ever been identified either since she is such a pleaser, she doesn't rock the boat, and is very good at blending in socially.
I've said it before - one would think educators would know the signs and be able to at least have a clue about what is going on. In our situations, I don't know what more the school needed to see.