Put a Media Module on a page, and configure it with the following characteristics:

• "Display Container" property = false,
• "Edit module" permission = true for a user in a role other than administrator

When a user of that role logs in and puts the page into Edit mode, no edit menu is displayed, therefore it is not possible for them to change the content of the module.

The module behaves correctly when the user is in role administrator, showing the edit menu even when the module container is suppressed.

this is by design, edit module permission for non-page-editors require a container with edit link to be usable.

This is not specific to Media module, but affects all modules.

Wll, np, and I don't like this design either as not very user friendly. We hopefully get the chance of improvements in future versions.