WordPress Version 4.9.8 CiviCRM Version 5.5.1
Giving permission to user 'edit own contact' will be able to update his own details. You can check permission by navigating to CiviCRM >> Administer >> Users and Permissions >> Permissions (Access Control) and then click on Wordpress access control
To View your own contact dashboard
1) Add WordPress Plugin Members
2) Go to individual Role's permissions page and then tick Access Civicrm.
3) Visit Wordpress Access Control under Users and Permissions as mentioned in answer given above and select following permissions.
- View my contacts
- Edit my contacts
- Access my contact dashboard