Two related questions:
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.
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.
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.
Because they are separate attacks, and not a single splash damage area, the answer to both of these questions is the same
Roll them separate.
This is because each target is targeted separately, and each attack is independent, so both the attack and damage rolls are separate rolls.
(Preface: since making this answer, I've revised my thinking and have added a separate answer that I believe to be more accurate; since its conclusion is different to this accepted one, I've left this one as-is.)
Purely RAW Answer*: Attack rolls are unclear, but damage rolls should be a single roll for all targets (including Witch Bolt's followup actions for the 1d12 damage per round).
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.
Twinned Witch Bolt is still a single spell instance, you've just changed its targeting mode from 1 to 2 targets. As such, it's one spell dealing damage to two targets at the same time (assuming it hits both), and with no more specific rule to beat out the above general one, it should apply. If I'm missing a more specific ruling that says one unique damage roll per attack roll, some or all of this may not apply, but I've not spotted such a precise statement in a quick lookthrough.
As for attack rolls, I can't find a specific RAW ruling on that. I think all AOE things with attack rolls either have an initial single attack roll followed by an AOE save around the target, or are spells/effects that have a specific ruling on attack rolls built into itself such as Eldritch Blast (separate per beam) and Battle Master Fighter's Sweeping Attack (use the original attack roll for the second target).
As such, there's no 'general' ruling on multi-target attack rolls I can find, as twinned spell seems to be the only way to actually cause one that doesn't have its own specific ruling built in.
*The RAW interpretation of this comes across as a bit weird, especially since there's a gap in the rules where there's no general ruling on AOE attack rolls I can find. I certainly wouldn't have a problem with a 'homebrewed' ruling to fill the gap, but I wanted to provide everything in the RAW that I could find.
Two independent attack rolls, if applicable. Two independent damage rolls, if applicable. Two independent saving throws, if applicable.
You cannot twin AOE spells. The spell must only target one creature to be twinned (PHB P.102)
