In Orthodox cultures during ancient times, the betrothal would many times be held in the Church as a service seperate from the actual wedding service. This betrothal was considered binding, and was basically a pledge of marriage so strong that to break it was considered a divorce. Now, it is important to remember that during these times, most marriages were arranged by the parents. Whatever the feelings of the kids (and they were kids, normally under 16), the parents were making a contract of sorts that involved social and political connections, sometimes large amounts of property, and so forth. The betrothal was as much a contract between parents as an agreement between the would-be bride and groom, who were usually too young to make such important decisions anyway. So there would be a betrothal, and then the kids might not marry for a year, or six years, or who knows when. This probably isn't something you'd read in Fr. John Meyendorff's or Fr. John Mack's fine books on marriage, but this does seem to be the way things happened.
As we get closer to the modern age, things changed a bit. The Betrothal service started getting performed together with the wedding service. Eventually, they got to the point where they were mostly done together, one right after the other, which is what is common today. At the same time, the age of getting married was increasing, and less and less marriages were being arranged. Obviously in America, much of this stuff isn't relevant to us.
Marriage--like all Sacraments except for Confession--is a public event, so it could be argued that the engagement should be public. However, since most Orthodox do a betrothal service before the crowning anyway, I don't think it's necessary. Of course, if you wanted
to, you could talk to the priest (and maybe talk to the bishop). I'm not sure about the intricacies of the Antiochian practice in particular (even though I was married in an Antiochian Church). I'd say, just ask the priest about it. Also, talk to older ladies and gentlemen in your parish; you will probably both benefit immensely from such talks