When a a post has been saved as a draft, there are multiple options for next steps. You can Post Now (without review), you can review the post with Edit/Schedule and you can delete the post.
From the Pending Drafts stream, click on the 3-dot menu.
If the post saved as a draft is ready to be published without any review, click Post Now. The post will immediately publish to selected account(s).
If the post needs review or changes are needed, click Edit/Schedule. This will open the Post Composer so you can review/edit the post before scheduling or posting now.
The user who publishes or schedules the the post saved as a draft will be the author of the post, regardless of the user who originally created the the post.