A question (big spoiler!)

Well, I think I answered my own question: probably after a soldier agrees to cooperate to that great a degree he would be kept separate from the other prisoners.

One of the conditions of the deal Basehart (not Widmark :-) ) made with Kim was to keep the arrangement a secret from the other prisoners. Probably Basehart believed that the men wouldn't be able to hide the fact that they knew the real story - a reasonable fear - and so didn't tell them, at great personal cost to himself. It was a brave thing to do, although of course also a betrayal of his country. The film posits an interesting and difficult moral dilemma.