Media Insert (Media Upload) doesn't work on author role in Front-End

Media Insert (Media Upload) doesn't work on author role in Front-End

I have a problem.
In my front-end I have a Write a Post area, where users can write a post,
and add their own images.
I have inserted the media upload with wp_enqueue_media();
With my current role: administrator I can successfully upload an image.
But a normal user (With Role: Author) cannot upload any images. After
selecting the image to upload, or drag&drop it, the image uploads but at
the end it gives: "An error occurred in the upload. Please try again
later."
I have no JavaScript errors, and the POST to async-upload.php returns a
200 OK but has no RESPONSE.
From the back-end both Administrator and Author works.
From the front-end only Administrator works. :(
Any ideeas?