2 Message icons on iPad doc

Hi, Jerri101. Disabling Handoff doesn't affect messages appearing on each device. If you're signed into iMessage and have the Continuity feature enabled (As described here: Use Continuity to connect your Mac, iPhone, iPad, iPod touch, and Apple Watch) then messages would appear on each device.

Take care!

Hi, jerri101, and thanks for reaching out to us on the Apple Support Communities. We understand that you're having concerns about what you're seeing on your iPad dock. We'll be glad to provide some insight.

From what you've described, it seems like you're referring to a feature called "Handoff". For more information on this feature, how it works, and how it's enabled; check out this article: Use Handoff to continue tasks on your other devices

We hope this helps you out.

Cheers!