Switch display and Magic Mouse & Keyboard between laptops?

I'd like a way to have my Magic Mouse, Magic Keyboard, and a display of some kind (Studio Display, ideally, or a non-Apple display if necessary) work with a personal laptop and a work laptop, and be able to switch between one laptop and another with the push of a button, perhaps something like a KVM switch. Is that currently possible? If so, how? Or do I have to manually re-pair the mouse and keyboard every time I switch laptops?