First of all, you need an MX record in your public DNS (mydomain.it)
Secondly, you need to set a custom helo for your mailservers public name (eg mail.mydomain.it)
In order to prevent your mail to end up in the recipients spam box, you need to configure an SPF record in your public DNS.
You will need to teel your mailserver to accept mail for mydomain.net and create aliasses for the public mailboxes and deliver that mail to the local users.
Anny issues, shoot … I have this working flawlessly, but it can be a bit tricky to set up.
Additional things you need to do afterwards:
Configure the mail property of your AD accounts, assuming you are using AD.
Perhaps this holds true for OpenLDAP as well, unsure.
In SOGo, if using that, go to the mail settings an alter the listed email to the public one for each user.