Edit the calendar part, and check the following two items:
  1. Uncategorized Event Management Rights
  2. Event Categories

In Uncategorized Event Management Rights, you can select which user roles have access to add/edit/delete uncategorized calendar entries.

In Event Categories, you may find a listing of various different event categories available, such as Birthdays, Events, etc. Here you can click the Assign rights to this category button (it looks like two people or a key icon depending on your version) and review which user roles have rights to access events assigned those categories.

If a user is having difficulty, check the above settings and confirm the user belongs to an appropriate role.