From what I understand, the laws of "bechor behemah tehorah" (kosher first-born animal) are still in effect today, both in Israel and outside. To avoid problems with this, we sell a share of the mother (I've always heard it explained as 'her ear') to a non-Jew, and therefore the firstborn offspring is permitted. Does that mean that the non-Jew who bought part of the mother owns a share in the offspring? Also, in today's huge meat-packing plants, does that end up being thousands of dollars a year? Does anyone supervise this today (such as the rabbanut in Israel)?
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Indeed the laws of firstborn animals (bechor) apply today and the Shulchan Aruch (YD 320:6) rules that we should avoid creating bechorot because the laws are very complex and people will likely make mistakes, AND because we can't bring them as korbanot nowadays anyway. He suggests that you sell to a gentile a part of the pregnant mother to make you a partner with the gentile, thereby exempting the embryo from the laws of bechor.
In terms of your question, it seems that we only sell part of the mother and not part of the child as the child is a Davar SheLo Ba LeOlam, a thing which his not yet entered the world about which it is impossible to make a kinyan on (formal transfer of ownership). Thus the child is wholly owned by the Jew.
In terms of the size of the portion sold to the gentile, the Shulchan Aruch rules two paragraphs before that it must be something that if it were cut off the animal would be considered blemished (mum). An ear is small and serves this function, so practically we often sell the ear. I suppose if the gentile would keep track of the animal and come to the slaughterhouse and claim his ear, they would give it to him. Practically, no one does this. It's much like selling chametz in that regard. For the gentile, it's a one time deal, and then he leaves and forgets about it.
I am under the impression that the religious kibbutzim in Israel arrange for this themselves, but have no source. (Most if not all kosher meat produced in the USA is from gentiles' herds.)