I have written this in the past, so you may have heard me before.
We used to live in Kazakhstan. The boxes were given out there. (operation christmas child)
The boxes were given to local churches to distribute. They were not given out at Christmas (actually the idea of christmas presents is pretty much a western construct and not all western countries either. My dh is dutch christian, they don't do gifts at christmas. I lived in Indonesia, christians there don't do gifts at christmas, Kazah and Russian christians - protestant evangelical, catholic or orthodox christians - don't do gifts at Christmas)
anyway, the churches do not give them out at christmas. Often at New Year (gift giving time in Kz) Often saved and used through the year.
Thye were given out in several ways, some churches used all ways, some one way or another. They were given out to children in the church; They were given out to all local children in the village/neighbourhood, with a message of greeting from the church; they were given out to local families for them to give to their children; the local children were invited to a party and given free food and a box, and a book or an invitation to church; they were put in the present sack of the equivalent of father christmas, who comes at new year.
So yes they are definitely given out by and through the church. In most cases I knew of they were given out to everyone in an area, the church was used as the agent to distribute them. The distribution was definitely with a christian message (like a leaflet) but was usually given to everyone without prejudice. This does also mean that they are given to all children not just the poor ones.
I have mixed feelings about them. Our school has sent home a leaflet, not sure if I will do them or not.