No, it does not. Please do a little research before continuing to spread misinformation. Apple servers are required for CSAM detection to occur, so if you don't use iCloud Photo Library, it doesn't work.
You are incorrect. I've read significant amounts of both their technical documentation as well as other "expert" analysis.
The CSAM hashes are on your phone. The system matches them with hashes on your phone. That happens no matter what. They don't leave the device unless iCloud Photos is enabled, but the hash matching happens on your device regardless.