Can I have Screen Sharing use something like ssh-keygen, so I don't have to type my password every time? I can use ssh-keygen to set up accounts on my two Macs so when I type "ssh me@my-other-mac" from here, it doesn't need me to type my password again every time I connect.

Is there a way to set up Screen Sharing similarly?
No, I don't think so. Screen sharing uses VNC which is a different protocol than SSH. SSH allows you to use keys which is the mechanism where you can not use typed passwords.