Mishnah 3 in the third chapter of Maseches Challah deals with the connection between the obligation of challah and ownership of the dough. The starting point is well known: only the dough of a Jew is obligated in challah, while the dough of a non-Jew is exempt.
A non-Jew who gives dough to a Jew to prepare for him:
The non-Jew hands the ingredients over to a Jew, and they remain in the non-Jew's ownership, except that the Jew is the one who actually works the dough. Such dough is exempt from challah, since the obligation of challah does not apply to the dough of a non-Jew. Even though the Jew kneads the dough with his own hands, working the dough is not sufficient reason to create an obligation of challah: the obligation depends on ownership, and since the ownership belongs to a non-Jew, it is exempt.
A non-Jew who gave the dough to a Jew as a gift:
The Mishnah continues with a case in which the non-Jew gave the Jew the dough, or its ingredients, as a gift, so that ownership passed from the non-Jew to the Jew. Here we must distinguish between two points in time, since the moment of gilgul, the point at which the ingredients become dough, is the moment the obligation of challah begins:
He gave him the ingredients before they became dough: it is obligated in challah, because the gilgul of the dough was done by the Jew, after he had already received the gift into his own possession as his own.
He gave him the dough after it had already become dough: it is exempt from challah, because at the time of gilgul, the moment at which dough becomes obligated in challah, the dough was still in the hands of the non-Jew and in his ownership.
From here we derive the practical halachah: one who buys frozen dough in a store does not separate challah from it, since the gilgul of the dough took place before the purchase, at a time when the dough was owned by a non-Jew.
Dough in partnership with a non-Jew:
The Mishnah continues with a similar case: a Jew who mixes dough in partnership with a non-Jew, so that part of the dough belongs to the Jew and part belongs to the non-Jew. The ruling depends on the amount in the Jew's portion:
The Jew's portion does not contain the measure requiring challah (five quarters of a kav): it is exempt from challah, since there is no challah-requiring measure here under Jewish ownership.
The Jew's portion does contain the measure requiring challah: it is obligated in challah.
In summary: in this Mishnah we learned that the obligation of challah is determined by ownership of the dough, and not by who labored in working it: dough of a non-Jew kneaded by a Jew is exempt; a non-Jew who gave the ingredients as a gift before gilgul creates an obligation, and if he gave the dough after gilgul it is exempt; and in dough owned in partnership, the obligation depends on whether the Jew's portion contains the measure requiring challah.