Enable Publishing Workflow
The Publishing Workflow let's you save the web part on draft, submit it for review and publish it. With this option enabled you Save as Draft, Send for Review or Publish.
Insert the emails of the users you want to give permissions to publish content that's in draft or was submitted to review.
If a web part content is save as draft or submitted to review it will only appear in edit mode.
Web Part content in draft will only appear to the person who created the content.