The original manuscript is not extant here for “and the(ir) privileges”. Spacing between extant fragments is sufficiently large that 𝓞 could have read with either their or the. Since both the 1830 edition and the printer’s manuscript read their, 𝓞 also probably read the same. The editors for the 1920 LDS edition emended “their privileges” to “the privileges” since this is what we expect in English.
The word their occurs five other times in this infinitive clause, which strongly suggests that Oliver Cowdery accidentally wrote their in 𝓞 instead of the correct the. (Another possibility is that Joseph Smith himself mistakenly dictated their here.) There is considerable evidence that Oliver sometimes wrote their instead of the in anticipation of a following their. Here are six cases of this error on Oliver’s part, although all but the first one are momentary errors and were for the most part corrected by Oliver almost immediately:
In fact, the last example shows precisely what is proposed to have happened here in 3 Nephi 2:12, namely, the replacement of the correct “the privileges” with “their privileges” (although only momentarily in Alma 51:6).
There are no other examples in the text of the kind of redundancy manifested here in 3 Nephi 2:12, namely, “their X of their Y”. In fact, in the last three of the errors listed above, the correct text read “the X of their Y”, but Oliver Cowdery initially wrote the text in 𝓟 as “their X of their Y”. Similarly, there are no related examples in the entire text like “his X of his Y” or similar redundancies involving other possessive pronouns such as my, our, or your. Interestingly, there is one case in 𝓞 where Oliver initially wrote “my X of my Y”, but almost immediately he replaced the anticipatory my with the correct determiner, in this case the indefinite article a:
In other words, the unique redundancy of “their privileges of their church” (the earliest reading in 3 Nephi 2:12) is very likely an error for “the privileges of their church”. The critical text will therefore accept the 1920 LDS emendation as the reading of the original text for this passage.
Summary: Accept in 3 Nephi 2:12 the 1920 LDS emendation that replaced their with the, giving “the privileges of their church”; the earliest reading, “their privileges of their church”, is apparently an error that entered 𝓞 when Joseph Smith dictated the text to Oliver Cowdery, probably in anticipation of the following their in “of their church”; there are four other instances of their in this infinitive clause that may have also influenced the replacement of the with their.