Up to D&D
As I covered in on multiple attacks, I don’t think multiple “to hit” rolls in one round are a good way to model fighting with two weapons in D&D.
I’m replacing most of the musings that used to be here with a link to how I am currently handling it.
Originally for damage the attack roll determined which weapon’s damage to roll: Even, primary hand; odd, secondary hand. But I decided to simplify it and just allow the player to choose which weapon’s damage to use.
Some people like to give the two-weapon fighter the option to have either a +1 “to hit” or a +1 to AC. If I were to do that, I’d want to up the AC adjustment for shields to +2.
WSmith from Dragonsfoot offered this option: Make one “to hit” roll. If successful, roll the damage for both weapons. Drop the lower damage roll.
What if one of the weapons is magical? Roll “to hit” & the damage for both weapons simultaneously. If the damage for the magical weapon comes out higher than the damage for the other weapon, add its bonus to the “to hit” roll.