It's not the blood decomposing the Hydrogen peroxide(H2O2). It is H2O2 which releases oxygen and water when interacting with hemoglobin.
As H2O2 is a oxidizing agent they oxidize certain enzymes in the cells which are responsible for cellular respiration thus killing bacteria.
Since our cells are equipped with some anti oxidants our cells can withstand some considerable amount of H2O2 in blood.
