I have a Role with Basic Access with Sharing, and another with Full Administration. When I use the Full Administration Role, I can save Personal Reports, but with the Basic Access role, the Save button is missing from the page. Is that a glitch, or is this something that can be configured?
Regardless of the rolls name, everything comes down to how the permissions are set up.
To access permissions, you'll need to be on an administrator account. Once you are, go to your home page and click settings. Once in the settings hub, click on roles. There you will be bale to view all of your roles.
Click on the role with basic access with sharing. You should now see a list of all of the tables in your application below the app access level setting. You will need to configure each table accordingly to how you would like the user to be able to interact with it. If you would like your user to save a record on that table, he will need modify permissions and view permissions. You will need to set that tables permissions to all records, custom rule (and set up logic to records he can modify), or records that are owned by him. Along with view/modify permissions if you would like the user to add/delete records, the checkbox will need to be checked as well.
Hopefully this has helped you.