It all depends on the individual- what their BMR is and other elements of their health and history. For some small people 1200 is not that far from where their normal BMR rests and therefore is an appropriate place to lose weight. For others, they have a much higher BMR, perhaps because they are taller or weight a great deal more, their body needs a lot more just to stay alive from day to day. For these individuals, 1200 is way too low.
Important to distinguish between 1200 gross and 1200 net. I am on a 1200 cal diet right now but it's 1200 net; I definitely eat more when I exercise. If I run for 30 minutes that will burn a good 300+ calories, so I will ensure I make up for that deficit by eating a good number of high protein and calorie foods to get my net closer to 1200.
It's possible to get in all the nutrients required at 1200, but it definitely takes planning. With a very limited amount of calories, one has to focus on superfoods (high nutrient foods), high protein foods (eggs, nuts, etc) and make sure those get in first. No room to eat white bread or a bagel if all it is going to offer you is carbs
