Yes, it’s possible. Technically the Mass is still open to all, but of course if only the invitees know the time and place that in itself restricts the attendees.
Depending on the parish, First Communion celebrations vary. In my current parish (small) it’s done right in the normal Sunday Mass. We only had 8 first communicants in our tiny little parish.
In my old parish, in a major metro city, we had >200 first communicants ever year. Therefore, they held FC on a couple of Saturday afternoons at special Masses for logistical reasons-- first b/c it would extend the normal Sunday Mass too much and also b/c with the extra family members all at attendance at the regular Mass it would push seating past capacity or require the families to limit their attendees.
So, those could be viewed as “private” in a sense. Anyone could go, but you’d have to know about it.
My friend, who’s Uncle is a priest and is her godfather, has done all the Sacraments for their family in what you could call a “private” ceremony. The baptisms, FCs, etc, are in his parish at a special time/day.
A wedding is the same way, one is typically invited to it-- but the Mass is actually open to anyone who wants to go.