By default, students may only read announcements that have been posted by instructors and other instructor-type roles. But permissions can be changed so that students may create, edit, delete, and even access draft versions of the announcements not yet published.
Select the Announcements tool from the Tool Menu of your site.
In the illustration above, students have been given access to create an announcement and to edit or delete an announcement that they created themselves. With these permissions they can not edit or delete the announcements created by others. (Yellow highlighting has been added for emphasis).
Note: You may see different roles listed depending on the roles which exist in your site. A role must be present in the site in order for you to modify its permissions. For instance, in a project site, you will see role for Access (i.e. students/participants) and Maintain (i.e. site owners/instructors).