SOLVED! I had this issue too, and it's one of several ways that Contact form 7 can break. In this instance, it's a javascript conflict, perhaps with a plugin or the theme itself, and that's keeping the user from seeing the success message, even as the admin gets the note. To solve it, add the following function to your theme's functions.php file:
// De-register jQuery from Contact Form 7
add_action( 'wp_print_scripts', 'my_deregister_javascript', 100 );
function my_deregister_javascript() {
wp_deregister_script( 'contact-form-7' );
}
Worked like a charm for me, hope it helps somebody else. Feel free to contact me if you need help implementing.