So the problem is solved at last. After help with analysis by my colleagues responsible for local mailserver, we found out, the problem was username for SMTP authentification.
I got confused, since it wasn't smtpuser@localmailserver.com (I'm referring to example I made), but smtpuser@localdomainname.com.
So advice for others with similar problem, try to add testing account(one you are using for authentification) to your e-mail client(i.e. Thunderbird) and once that is allright, you have right credentials. Also try sending mail from that account elsewhere.