The module Node Embed does not work for a custom user role
In the Module Documentation for both CKEditor Integration and WYSIWYG Integration it says:

"Enable the default view 'ckeditor-node-embed' (if its not already enabled). This view provides the content of the 7plugin dialog box."

The access to this view is "Permission | Administer Content"

If your custom roles do not have this permission it won't work.

I altered the access to be by Role and selected roles I wanted to be able to use Node Embed and it then worked.