When twinning a spell that uses an attack roll, you roll two attack rolls and two separate damage rolls.
When twinning a spell that does not use an attack roll, you roll damage once.
The general rule for when a spell deals damage to multiple targets at the same time is as follows:
If a spell or other effect deals damage to more than one target at the same time, roll the damage once for all of them. For example, when a wizard casts fireball or a cleric casts flame strike, the spell’s damage is rolled once for all creatures caught in the blast.
[Basic Rules, Damage Rolls]
Twinning a spell causes the same spell instance to get a new target, so normally this would be the rule that applies when determining whether or not to roll damage multiple times. Something like a twinned Poison Spray would have a single damage roll that covers both targets as a result.
For spell attacks, however, there is a more specific rule that determines when you roll damage. (Emphasis mine)
Whether you're striking with a melee weapon, firing a weapon at range, or making an attack roll as part of a spell, an attack has a simple structure.
Choose a target. Pick a target within your attack's range: a creature, an object, or a location.
Determine modifiers. The DM determines whether the target has cover and whether you have advantage or disadvantage against the target. In addition, spells, special abilities, and other effects can apply penalties or bonuses to your attack roll.
Resolve the attack. You make the attack roll. On a hit, you roll damage, unless the particular attack has rules that specify otherwise. Some attacks cause special effects in addition to or instead of damage.
[Basic Rules, Making an Attack]
The rules for making an attack are specific that when you make the attack roll, if you hit, you roll damage. Since you're making a separate attack roll for each target, you roll damage separately for each hit as per step 3 in the quoted section. (You could also argue that these separate attack rolls mean you're not dealing damage to the two targets at the same time, so the earlier rule on hitting multiple targets at the same time wouldn't apply anyways)
Worth noting that while the initial attack of a twinned Witch Bolt would be two attack rolls and two damage rolls, Witch Bolt's optional damage on the following turns would be one roll for both targets since it no longer requires you to make an attack roll to deal the followup damage.