What Permission to view a user's own Profile/Dashboard
Dashboard &amp; Permissions

If you want to give the user ability to see information on themself, the permission seems to be CiviCRM: access Contact Dashboard. (I expected CiviCRM: view my contact to be a requirement also, but it doesn't seem to be necessary for a contact to view their own dashboard.)

If you also want to give the ability to edit the logged in user via the contact dashboard, the permission is CiviCRM: edit my contact.

Permission to access to other CiviCRM components may dictate which Dashboard elements appear on the user's dashboard.

Here's CiviCRM's documentation on permissions.