In the middle of this passage—in fact, within the same sentence—the earliest text switches from the first person plural to the third person plural. The first-person usage begins in verse 12 (“to be judged according to our works”) and continues into verse 15 (“we must come forth”). The rest of verse 15 is in the third person plural, beginning with the conjoined predicate “and acknowledge to their everlasting shame”. In his editing for the 1837 edition, Joseph Smith emended the third person their to the first person our, thus making the following pronominal form agree with the sentence-initial subject pronoun we (“we must come forth … and acknowledge to our everlasting shame”). Later in the sentence, the text refers to the salvation of mankind in the third person, not the first person: “and that he is merciful unto the children of men and that he hath all power to save every man that believeth on his name”.
As noted under Alma 11:41, there is no evidence that scribe 2 of 𝓟 ever mixed up the number for pronouns. Nor is there any specific evidence that he ever mixed up the person (such as switching the first and third person plural pronouns). On the other hand, there is evidence that Oliver Cowdery occasionally switched the person for these plural pronouns, but only under the influence of the grammatical person for words in the preceding text:
It is therefore possible that in Alma 12:15 Oliver Cowdery mistakenly wrote their instead of our in 𝓞. However, in each of the three examples listed above, there was a nearby preceding form of the opposing person that seems to have prompted the error. No such preceding form can, however, be found here in Alma 12:15; that is, there is no third person plural pronoun in verse 15 that could have caused an original our to be replaced by their during the early transmission of the text. Perhaps one could argue that the immediately preceding sequence of third person singular pronominal forms, him and his, led our to be replaced by the third person their (“and stand before him in his glory and in his power and in his might and majesty and dominion”).
There are other instances in the text of such sudden shifts from first to third person, including one noted example in Alma 56:52–53 involving an apparent shift from a direct quote to an indirect one, with concomitant shifting from first person to third person (see the discussion there as well as under Alma 5:5). The critical text will accept here in Alma 12:15 the earliest reading, “to their everlasting shame”, despite its difficulty. There seems to be little reason for the their except that the original text read that way, although David Calabro (personal communication) points out that the their might be Alma’s own attempt to distance himself from having “everlasting shame”. Even so, the preceding verses (13–14) use the inclusive first person plural to refer to what will happen at the day of judgment “if we have hardened our hearts against the word”. Calabro suggests that the shift to the third person occurred in verse 15 because of its considerable distance from that first person if-clause at the beginning of verse 13.
Summary: Restore in Alma 12:15 the occurrence of their in the earliest reading (“and acknowledge to their everlasting shame that … ”); this difficult third person usage seems to have been the reading of the original text.