I think I've figured out something using the Reveal plugin, which tells me which template file is being used on each page.
The pages that have a sidebar are the ones I've created myself like this page. They seem to be using the /twentyeleven/sidebar-page.php template.
The pages that Events Manager auto generates for Events, Locations and Categories are using the /twentyeleven/page.php template. These pages don't have sidebars like this page.
So, I tried modifying the page.php template and adding a <?php get_sidebar() ?> function but it wouldn't properly position on the right side no matter where I placed it in page.php.
It feels like there is something wrong with my installation of Events Manager or WordPress. I think the single-event.php template I made is being fed into the page.php and that is causing problems.
I've been working on this for 24 hours and sadly am not any closer to the solution.
Does anybody have any ideas for me to try?