Zotero lets you share references using a feature called "Groups." Groups are a good way to share references with a class, work closely with research collaborators, and keep track of conversations happening in your field.
Groups are separate from your personal library and are created online, rather than in the software you downloaded to your computer. You will be able to see references in a Group locally, but permissions are set online, in the Group settings. You can drag and drop items from your library to a Group and it will make a copy of the item.
Refer to the full Zotero documentation for details on permissions and adding members to a Group.