Can your character maintain concentration on two spells?
If concentration is broken, do both spells fall, or are separate checks made on each?
When you cast a spell ... you can ... target a second creature in range with the same spell (1 sorcery point if the spell is a cantrip) (PHB 102)
Note that the text says that it is the same spell. You can't concentrate on two spells as you say, the first spell's concentration would end because you would need to concentrate on the second spell.
Instead, treat a twinned spell as if casting a single-target spell like Hold Person that has the option to cast at a higher level to target an additional creature.
As for the concentration breaking: Since you are only concentrating on one spell, only one check is needed to fail for the spell to end on both targets.
Witch Bolt has the following mechanic:
... The spell ends if you use your action to do anything else. The spell also ends if the target is ever outside the spell's range or if it has total cover from you. (PHB 289)
It's your turn, you cast Witch Bolt and spend sorcery points to use Twinned Spell. You cast Witch Bolt only once but you get to target two creatures.
After making two attack rolls, let's assume you hit both creatures (because if only one hits then you simply resolve the spell like normal). Now, both creatures take 1d12 lightning damage and on your subsequent turns, you can use your action to automatically deal another 1d12 lightning damage to both creatures again.
The spell ends on ONE creature if the creature moves out of range or has total cover from you.
The spell ends on BOTH creatures if you use your action to do anything else except dealing the 1d12 automatic lightning damage OR you are hit and lose concentration.
Twin Spell doesn't cast two spells. It allows a spell which targets a single target to target two targets instead. The ability doesn't modify the number of spells being cast. It simply modifies the number of targets the spell you're casting has. It's otherwise unchanged. Since it's only one spell, if you lose or stop concentration on the spell, the spell ends for both targets.