AJAX failing in multisite with dedicated domains

Hi all,
in a multisite installation with dedicated domains for my two sites (example.com / example.de) I have set up an AJAX call using the regular WP procedure. I use a handler action in functions.php like add_action( 'wp_ajax_nopriv_some_task', 'do_my_task'); and so on which works perfectly on the site with blog ID 1 (example.com). But switching to the other site (blog ID 2, example.de), the JS callback function receives an empty response, the handler action apparently never gets called
I’m using the Multilingual Press plugin which provides the functionality for the two domains, and I’m not at all sure if the problem couldn’t be possibly related to this plug. Maybe I’m running into some cross domain issue?
Anyways, I combed through the web for a couple of hours now and couldn’t find any suggestion that seems to apply to this case.
Any ideas?
Thanks so much in advance!
Regards,
Soela