Users can update the appearance of the data table through changing the column formatting, adding or hiding columns, or adding and modifying the data itself. In order to make these changes, you will need permissions to create and publish working copies of the dataset. You can see more about roles and permissions in the support article, Understanding User Roles.
Please note, in order to make any changes to a dataset's appearance, you will first need to create a working copy. The options available to format your dataset are only available in working copies of the dataset.
If you are working on a derived view (i.e. classic chart, filtered view, etc) you will not need a working copy, and changes can be made through the Edit menu.