The software to move screen/laptop to apple TV screen

Firstly, in order for AppleTV to see the iTunes Library a user must be logged in and running iTunes. If both of these are not present then AppleTV won't work.

Since they are on the same physical Mac might I recommend combining the iTunes Media folder and placing it in the /Users/Shared/ directory. As all users have access to the Shared directory the actual media files wouldn't be duplicated and they can each maintain their own iTunes Library. After moving the iTunes Media folder to the Shared folder you will need to go into iTunes Preferences for each user and under the Advanced tab select the new iTunes Media folder location.

Okay, after dealing with this for quite some time, I finally found a solution that worked for me. I figured I'd come and share for others as well.

In order to resolve the choppy playback, I ultimately went in to Settings -> General -> Reset -> Reset All Settings. After resetting, the device downloaded the very latest OS for the Apple TV, and after everything was done, the Airplay quality between the MacBook and AppleTV is actually pretty decent.

In testing, I was able to mirror my display without any noticeable choppiness. I was also able to extend my desktop on to the TV, and this worked surprisingly well. Even playing YouTube videos was of decent quality, with almost zero choppiness during playback.

What I have not tried yet... and will update this answer later on, is reconnecting my photo streams. As it's common to have 1000+ photos in my shared streams, I'm concerned that this may have slowed the device down (just a guess, honestly). I'll do some more testing today, mirroring as is, then adding the photo streams and comparing, to see if that causes any issue. If it does, I can always reset again.