Electrochemical water splitting liberates H2 and O2, but note that if H2 is produced in working electrode then O2 is liberated from counter electrode and vice versa. Next, you concentrate on the thermodynamic potential which is different for H+/H2 (0 V vs NHE) and OH-/O2 (1.23 V vs NHE). Moreover, if you see the mechanism of H2 evolution where H+ adsorption on the catalyst surface is the first step, known as Volmer step, followed by Heyrovsky or Tafel steps. A good HER catalyst always binds H+ very fast and release the product. In contrary, OER steps depend on the OH- adsorption on the surface. So a good HER catalyst may not effect towards OH- adsorption or vice versa. One more problem associated with OER is at high overpotential the catalyst often gets oxidized (oxygen binds to the metal surface strongly). It's not a fact that good HER catalyst cannot show OER, some oxide based bifunctional eletrocatalysts exhibit good HER and OER.
I've just replied your comments to another question. You write again that "you need to know the CB and VB level values for your materials." This question is clearly formulated. "the CB and VB level values" are absolutely unrelated to the original question. I had again to downvote your answer. "Also literature study will help you further." I absolutely agree with this and personally I always follow this rule.