By default, Admins and Editors can post Announcements. A Blogger or Contributor can post Announcements if they are given permission to (but in the case of a contributor, the admin approving the post can choose to turn the announcement on before publishing if the contributor does not have permission to). Permissions are managed through the blog settings on the front-end by clicking on the pencil icon next to the permission name.
Blogs - Boost Blog Engagement With Following and Announcements FAQ - Who Can Send Announcements?
