We don't know for sure whether the Centurions actually had all the same kit that the Legionaries had. We can only rely on depictions that portray the Centurions with Transverse Crests wearing armour, sword and dagger, and holding the vinewood cane.
Battle however is a completely different realm.
The Centurion had to have the freedom to directly command and control his troops in reaction to both signals received from the Cohort Commander or Legate as well as react to the changing scope of the pitched battle. Logic dictates that he would have to move quickly to points in the line of his Century. A shield would encumber his movement. He carried his weapons really for his own self protection as the Century itself was his weapon in the truest sense. Second school of thought - that the Centurion carried both shield and weapons on the battlefield.