We dont rely on eid echad neeman beisurin in our local stores, so why do we rely on the owners of most schools and camps that everything is kosher?
Yeshiva (based on my kid's elementary and high school):
The elementary school had an optional lunch program. The yeshiva publicized exactly who the caterer was - a well established neighborhood caterer supervised by two Va'adei Harabanim. (I won't delve into the politics of why you need more than one, here.) If that wasn't sufficient for parents, the yeshiva dictated which supervisions of food kids were allowed to bring from home. No, it was not fool proof, as no one was checking a mommy-made bologna and ketchup on white bread sandwich. But, then, again, students with their own lunches had to eat in the classroom rather than the lunch room to prevent possible kashrut problems.
So, other than the lunch program, there was no way to enforce a specific kashrut policy other than to recommend guidelines.
Between my nieces, nephews and my own kids attending 3 NY State summer camps over the past approx. 30 years, each camp had a mashgiach temidi, from what I recall. While the camps did not publish where they got their food from, they had no problem revealing this info to anyone who asked.
I worked as an assistant head waiter in Camp Massad in the 80's. When I was there, there was no mashgiach on premises, but, someone came by once a week to inspect, and we did not know when he would come. Additionally, Rabbi Eliach (Sr. - not his son Yotav) was one of a few rabbis who would almost constantly check things in the kitchen (and sometimes tell the chef how to cook things!)
@DanF makes a valid point about this being "a wrong general assumption". I cannot speak for all camps, but I've been working for a chassidic boys' camp for a number of years: they have a hashgacha from an outside agency that certifies their kitchen. While I don't work for the affiliated school all year, I do know that their kitchen (in Boro Park) is certified as well.