why can a right handed person not do with left hand, the same things as briskly as he does with his right hand? When brain can send a signal to the dominant hand, why can't it send to the non dominant hand?

As Tiger says, practice is probably a key element.

Muscle memory is very important.

Consider typing---I am definitely right handed and right side dominant; but I learned to type one summer at a business college when I was a teenager (good call Dad) and I can type 60 wpm reliably. 

My left hand seems to reach the appropriate keys as easily as my right.