Bump. The post above is from iOS 6 in 2015, but still relevant..
It's 2018 and my iOS 11.2.5 Apple Watch / iPhone X are
How do you turn Family Reminders off?
I had to turn off Reminders completely on my iPhone (now I see only Reminders I created on my iPhone -- on my iPhone -- not my Mac)
I recently added my wife to the "family". At the same time I start adding reminders through Siri on my Apple Watch
"Hey Siri.. remind me to take my ham sandwich to lunch tomorrow"
Wife sends text "why am I seeing this on my phone?" ("Take Ham Sandwich to work").
(Yes, I went into my phone and deleted all the family reminders one by one -- but Siri adds them back)
But, I don't want Siri telling my whole family about my upcoming proctology exam..