How do I enable screen recording for myself only on iPhone?

Start here, Record the screen on your iPhone, iPad, or iPod touch - Apple Support

Screen recording on iPhone has always been to only record your screen, never anyone else’s. Otherwise that would be a major privacy and security violation.

Swipe down from the top right corner of iPhone to access the Control Center. Tap the + in the top left corner. Tap Add a Control in the bottom. Tap Screen Recording to add it back.

Now whenever you want to screen record you tap that. It will start a 3 second countdown before recording, and to turn it off return to the Control Center and tap it once more. Saved footage will go to your library.

Pull down from the top right of the screen. Look in the top left corner for the + add the screen recorder icon from the Control Center