When I first messed with the editor I changed to dialogs for the reason that at the time it wasn't possible. Perhaps with a custom script but I'm not a coder myself. I would recommend to use dialogs instead because as far as I know it isn't possible.
Unfortunately Screen Buttons cannot be shown or hidden for just one player.
You could use a small dialog with a dialog button instead. These can be hidden, moved or changed in size for each player individually.
You should mess around with adding custom images on buttons. You can make your own looking dialogs like that using textures in the files or textures you import. In my own map I am working on creating a menu while hiding the actual dialog so you don't see that race stuff, but displaying images and buttons with images instead.
I was thinking of doing that, but now looking at some of the images in-game (like the ones used for Blizzards screen buttons), they often have multiple images combined together into a single image. How do we access only part of an image like Blizzard does?
Thanks to Arkless from The Helper, I've figured out a way to make Screen Buttons display for specific players. According to him, Screen Buttons are nothing more than Dialogs with the background hidden. So we should be able to use the Dialog - Show/Hide (which DOES allow for specific players/player groups). I tried it out, and managed to get it to work correctly!
So yes, it IS possible to display Screen Buttons to specific players, but you need to use the Dialog Show/Hide.