Salesforce: View Dashboard As

In addition to Create and Customize Dashboard permission, there are a few more permissions related to the dashboard, such as Create Dashboard Folders. But to control dashboard builder on "View Dashboard As", there are 2 permissions related to this.

1. Only Create and Customize Dashboard permission

This user will not see the "View Dashboard As" option under dashboard properties

2. With View My Team's Dashboards permission

This user will see "View Dashboard As" with options: me and another person only.

3. With Manage Dynamic Dashboard permission

This user will see "View Dashboard As" with options: me (dashboard creator), the dashboard viewer (login user) with the option to let the dashboard viewer choose user beneath them in the role hierarchy.

For "me", do not select anything.

4. With View My Team's Dashboards + Manage Dynamic Dashboard permission

This user has all option to configure the dashboard for "View Dashboard As"

However, for "another person", you can choose any user below you in the role hierarchy. You can choose any user in your organization to be a running user of the dashboard if only have View All Data permission.

Reference: 

• Dynamic Dashboards: Choose Who People View a Dashboard as in Lightning Experience