If both occupied the same positions then the waves would exactly cancel and there would be no radiation. If instead they are very close together, they would appear from a distance to be electrically neutral but their radiations would not entirely cancel. Then we have radiation from a neutral body. I think you are going in the right direction. Dig a little deeper.
People forget what happens in a teenager's life.
There is nothing that says they can't experience the same thing. I agree with you that no two things can occupy the same place at the same time but you haven't shown any reason. But as the waves pass they both reappear. If an equal charge of opposite polarity is accelerated in the same direction at the same rate, it radiates in exactly the same manner except in opposite phase. If both occupied the same positions then the waves would exactly cancel and there would be no radiation.