From Joomla 3.4, Joomla have introduced a new user permission for frontend module editing. This change means that your users will now need to be granted permission to inline edit modules.
To do this:
- Click "System" in the menu then Navigate to 'Manage' > click on 'site modules' and click on the 'Options' button, at the top righthand side of the screen. This will take you to the 'Module Manager Options' screen, seen below.
- Click on the user group you wish to allow to frontend edit (in my example, I have chosen 'Publisher') and under 'Frontend Editing' select 'Allowed' from the drop down list.
- Click on the 'Save' button.
Your users will now be able to inline edit modules.