Think about it - going through your everyday, how much of what you do is one handed or one legged?
That's why we do dumbbell work. Each side had to work independently of the other, and the stronger side can't help the weaker side.
Have your squats stalled? Try some single leg work. Again, it's good for strengthening one leg independently of the other and it's also really important in learnign to stabillize the pelvis. Signle leg work, while sometimes included in the workout as lunges or pistols, usually shows itself in the warm up since it's better done at a slow and deliberate pace.