CiviVolunteer permissions to allow registered (Drupal) users to view assignment details
Rowan...

First, I think you want to remove "Access CiviCRM" from all users, unless you want them to be able to access CiviCRM's back end. Ordinarily, you assign that to people who have access to database functions (entering and updating contacts, recording donations, etc) and not people who just need access to the contact dashboard.

I'm pretty sure that the permissions you need to assign to the Authenticated User role (or another role if that's more appropriate) are "View My Contact" and "Edit My Contact." Volunteer assignments are stored in CiviCRM as activities and the contact permission also gives you access to the activities associated with a contact.

There currently isn't a built-in UI for volunteers to update their assignments, although there has been work done for volunteers to report their hours worked. The lead developer, Frank Gomez, is out of the office this week. I'll make sure he sees this and can give you a better idea of when this will be released. On Drupal, it might be possible to create something with the Views and Webform CiviCRM modules.

Other clients have requested some type of Volunteer Dashboard functionality. Everyone's idea of what this entails is a bit different, though. I'll reach out to you separately to get some details about what you're looking for.