Dubble tap homebutton and see whats going on on my partners iPhone, how do i turn that off?
its
It sounds like you may be referring to Handoff. You can turn that off in Settings > General > Handoff & Suggested Apps.
More details about Handoff: Use Continuity to connect your iPhone, iPad, iPod touch, and Mac - Apple Support
You should also consider using separate Apple IDs for each person. Sharing an Apple ID is becoming less and less desirable as more and more things are synced. You can use separate IDs and still share selected items via Apple Family Sharing.