How do they decide which twin needs to be released in The Giver?

The Nurturers weigh both twins and release whichever one weighs less.

In Jonas’s community, people go out of their way to make sure no one is uncomfortable. Ironically, in a place where conformity is everything, being identical is not allowed.  The community carefully controls the birth rate and the physical features of its citizens, but apparently they cannot control whether there will be one child or two.  It seems that non-identical twins are acceptable, so they do not intervene until they know that the babies are identical.  Consider the conversation between Jonas’s mother and father.

Right now we're all preparing for a release we'll probably have to make very soon. There's a Birthmother who's expecting twin males next month."

"Oh, dear," Mother said, shaking her head. "If they're identical, I hope you're not the one assigned--" (Ch. 14)

From this we can assume that the twin will only be released if the babies are identical.  They are preparing for the release, so they seem to know that they are identical or that they might be.  Interestingly enough, even though no one in the community has any concept of death, releasing seems to be considered an unpleasant job.  Jonas’s mother tells his father she hopes he’s not the one to do it.

Jonas’s father explains the process that he will use to ensure that the right twin is released.

I'll have to select the one to be nurtured, and the one to be released. It's usually not hard, though. Usually it's just a matter of birthweight. We release the smaller of the two." (Ch. 14)

Why do they do this?  In this community, it is considered improper to not follow the rules.  Rules are incredibly important.  The community functions under a series of rules for almost everything, which are designed to maintain conformity.  Everyone follows the rules, and no one is ever uncomfortable because they know exactly what to do, and do exactly what they are told.

Jonas’s father does not seem to understand that he is killing the baby.

"Do you actually take it Elsewhere, Father?" Jonas asked.

"No, I just have to make the selection. I weigh them, … and then I get the smaller one all cleaned up and comfy.  Then I perform a small Ceremony of Release and--" He glanced down, grinning at Gabriel. "Then I wave bye-bye…" (Ch. 17)

Actually, Jonas’s father is taking one to Elsewhere.  That is what killing the twin does, since Elsewhere is a euphemism for death.  Yet Jonas’s father does not seem to think anything of it.  You can tell by the fact that he talks about waving bye-bye to the baby as if it is just leaving the room, and not dead.  The fact that he looks at Gabriel, who is constantly threatened with release, is even more disturbing.

When Jonas tells The Giver about the release, the old man says, "I wish they wouldn't do that," and Jonas tells him that it would be confusing to have two identical people (Ch. 19).  Apparently that simple fact, that someone might mistake one for the other, makes people uncomfortable.  They are uncomfortable enough to kill one of the babies to prevent it.

When Jonas sees the ceremony of release, which was videotaped, he is completely shocked.  The event is a turning point for his character.  This climactic event causes Jonas a great internal conflict.  He simply cannot believe what he sees, and from then on finds it impossible to exist within his community’s norms.  Instead, he and The Giver plan his escape.  They hope to return the memories to the people, and with them the ability to feel.  When you can feel emotions, you no longer cannot react to the murder of an innocent baby.