Apple Watch in Family mode, what control does watch wearer have?

If I activate an Apple Watch for my child using family mode on my phone, what control does my child have over the watch?

My child actually has an iPhone but with no cell coverage plan (Wi-Fi only), so they can’t link the watch to their phone to purchase cell coverage for the watch.

Can they install the Apple Watch app on their own iPhone to have some controls like sync their contacts and photos and custom watch faces and other apps (e.g. Strava with their own log in)?

Everything I find online and on Apple website describes what the “parent” can control, but not what the watch wearer can control.