You are allowed an extra baggage allowance or an extra 20kg (when they have a limit by weight, not by baggage #).
Most airlines have a limit of 3 for Business passengers, which means you can get a 4th free of charge. The 20 extra kg is only when an Airlines have a limit of (for example) 46Kg, it doesn't matter if you check in 2 or 3 bags, the limit can't got over that unless you pay a fee, on those occasions, is when the extra 20kg benefit apply.
With AA you can get 3x 70lbs in Y, or 4x 70lbs in J/F, but I'm not sure about others OW partners or if your companion will get the same benefit.