In separate saucepans over medium heat, cook the potatoes and carrots in water until each is tender. Drain, then cut into a ½-inch dice. In a large bowl, combine the beets, potatoes, carrots, pickles, scallions, and dill. Toss gently, season with salt and pepper, and add the eggs. Cover and refrigerate.
In a blender or food processor, combine the yogurt, mayonnaise, horseradish, lemon juice, and sugar and blend until smooth. Season with pepper.
Pour the dressing over the salad, toss to coat, and chill until served.