In general I think people test permissions by going to the permissions page and tweaking it until it works. That's an OK process, but can have unintended consequences (you may grant more or less than you wanted).
So, first, I suggest setting up test users that you can use to log in to your site with various different combinations of permissions. That's an effective way to test out your changes.
Second, set up a test site for this kind of experiment. Doing that kind of change on a test site will help you be more confident when you tell a user that you've fixed their permissions.
Often the README.txt or project page or documentation (if it exists) will tell you what permissions to use.
My final suggestion is that the one way to be really confident that a user can access a page is to look at the code. The
hook_menu of translation_overview.module and the
hook_permission together shows that users with "View translation overview assigments" should be able to see the path at
admin/content/translation-overview-assignments. However, there is also a permission for
'manage ' . check_plain($lang_code) . ' translation overview priorities' which is relevant. Note that there is a typo in the permission where it is missing the letter "n" in assinments.
It's possible that there is a bug in the module's permissions enforcement. The module's bug queue has a few issues that could be related.
Also, it's possible that the menu is broken and the person can access the page, but the link isn't presented. I would try visiting the page directly rather than using a link to see if access is the problem or navigation.