Honestly I would not jump straight to dairy issue. I feel like everyone is always screaming dairy issue (fwiw I do have a child with a legitimate dairy issue and he never has bloody stools) . If it is a 1 time occurrence I would not worry, however of it happens again I would go get it checked or at least talk to the dr or nurse and if the green poo and blood continues explore the possibility of a dairy issue.
Privacy? Who needs privacy