Most calendar servers, like Google Calendar, Exchange and iCloud sends an email to guests that they can respond to, with texts like "Albert has invited you to join Weekend BBQ, are you attending? yes/no/maybe". This function is completely separate from DigiCal, what we do is just saving an event on the server, most users on Android uses Google Calendar, and if the event has guests, the server will send out those emails automatically.
If you're using only a pop3 account and don't get guest invite emails, you're probably saving events to a calendar on the device itself. Local calendars generally can't send out invitations. You can register for a free gmail or outlook.com email to get hosted calendar services.
Invitations are normally sent out by the calendar server, but if they just disappear, I'm afraid that the Android sync adapter or the calendar server you're using doesn't support it for some reason.
We've confirmed that invitations works with an Outlook.com Exchange ActiveSync connection, but we know that certain non Microsoft Exchange compatible solutions don't work well with Android.