What is the correlation between electronegativity and the sharing of electrons?
Is there a specific equation that can give us how the bonding electron pair is shared?


No. The electronegativity scale is based on bond strength, not on partial charges. There are quantum-chemical methods to calculate the electron density in a molecule, and you can derive partial charges from that.


  For example in a C-O bond, the electronegativity difference is 1.2. does this mean that the Oxygen has 1.2 of the shared electrons and Carbon 0.8 (or some other numbers)?


No, the electronegativity difference does not allow you to make a quantitative statement like that. It is useful, however, to predict how a bond would hydrolyze. It also gives you an estimate about how polar a bond is.