Apple Watch messages and email

Yes you never said which watch it is,this is what you need for family setup;

• An Apple Watch Series 4 or later with cellular, or Apple Watch SE with cellular, with watchOS 7 or later.
• An iPhone 6s or later with iOS 14 or later for the initial watch setup.
• An Apple ID for yourself and one for the family member who will use the Apple Watch. Your Apple ID needs to have two-factor authentication turned on.
• A Family Sharing setup that includes the person who'll use the Apple Watch. You need to have the role of organiser or parent/guardian to set up an Apple Watch for a family member.

Is just the child using the watch or both of you?.

Set up an Apple Watch for a family member

Tap on the link I posted;

Set up an Apple Watch for a family member

Hi ohamydear,

We see in the subject you mentioned Apple Watch messages and emails, however the question mentions iPhone. If you're wanting to prevent messages and emails from coming to the device while the child is using the iPhone, you can enable Screen Time and use the downtime feature to set which apps the child is able to use. You may see other great options within Screen Time settings. For more information about Screen Time, check out this article: Use Screen Time on your iPhone, iPad, or iPod touch.

Here's another article that may interest you: Set up Screen Time for yourself on iPhone

Cheers.