Keyboard shortcut to switch screens on split view

Here is another partial solution. If you split your screen between two windows of the same program, e.g. Safari, then Cmd+` (backtick) will toggle between the two.

Caveats and Special Cases:

• If for some reason this does not work, you probably don't have this shortcut enabled. The answer at https://superuser.com/a/187072 explains how to do this.
• This was tested on High Sierra, but should work just the same on El Capitan, and so on.

I really hope someone has a better answer, but if not, Command+Tab works if the two last apps you've touched are the two apps being splitscreened.

Go to System Preferences > Keyboard > Shortcuts > Keyboard

Click the ^f4 and type your prefered shortcut. (I set mine to control tab)

It is possible

For Google Chrome for macOS users:

Hit Control + F4 to go the right screen.

Hit Control + Shift + F4 to go to the left screen.