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:

1. Navigate to 'Extensions' > click on 'Module Manager' 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.

2. 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.

3. Click on the 'Save' button.

Your users will now be able to inline edit modules.