How were potential missionaries identified, vetted, and called on missions? It appears that the local quorums of Seventy recommended and interviewed potential male missionaries.  Other than that, however, I mostly don?t know: the diaries tend to start when the missionary leaves home, with nary a word about the call or preparations.
In contrast to most of the other mission diarists I?ve encountered, Sister Amelia B Carling, who served in the Southwestern States Mission, 1901-1902, described the sequence of her “appointment” in detail.  I have transcribed her account below. 
While in Salt Lake City attending the General Conference in April 1901, I met Prest. James G. Duffin on the Tabernacle grounds on Sunday afternoon. I had met him the fall conference before, when Elders Meeks and Hoyt started on their missions, so I wanted to enquire how they were getting along, as he had just come from the Mission.
Prest. Duffin asked me ?how I would like to take a mission?? He state that there was a nice work opening up in his mission for two lady missionaries.
I told him I would very much like to go on a mission, that if I was qualified, nothing would give me more pleasure than being able to help in that great work.
Well of course I thought nothing of our little talk, any more than I wished that I was qualified and counted worthy to fill a mission, but I dismissed the thought from my mind as an impossibility.
The next week was the Utah Stake Conference, and Saturday morning after meeting, Minnie and I were walking down Center St. going down town to do our Saturday?s shopping.
Bro. Brimhall was standing on the sidewalk, and as we passed he says ?say Sister Carling I want to speak with you a minute.? so we stopped and he said, ?well what would you say if you were called on a mission, what would you say if they asked you to go?.
Well I said I would say I?d go if the [they?] wanted me to, and if my papa was willing, and would furnish the necessary means.
?Well,? Bro. Brimhall said, ?They want you for a mission, but it is not decided yet whether you will be called, but you are to see Bro. Reynolds the next time he comes down to give his lecture, and then the matter will be decided.? I was so shocked that I shook all afternoon.
The next Monday morning when Bro. Reynolds came I told him that Bro. Brimhall had told me I was to have a talk with him about a mission.
?Well? said he, ?we only want to know if you will be willing to go, as you are wanted for that mission?. I told him yes I was if I was wanted, and he said ?well that settles it, you will receive your appointment soon.?
I can hardly describe my feelings. I felt thankful, and sorry, a feeling, a mixture of pleasure and pain.
I had had for some time a longing to fill a mission, but never thought of such a thing as little ?me? being ?called on a mission.? That sounded too big for me. It seemed like it would be some one who was better prepared than I; but nevertheless, I felt that if the Lord had a work that I could do I was willing and anxious to do it.
Well I was tried in my feelings severly before I received my appointment. I wrote to Papa about it and found that he was in such financial embarisment that he was not able to furnish the necessary means so it was a little doubtful whether I should be able to go, on this account.
I felt that the call was from the Lord, and that it would break my heart to have to refuse.
I saw in this trial the fulfillment of Nephi?s words when he said that the Lord never calls on us to do anything except he will open the way for us to do it, if we are humble and put our trust in Him.
Well, satisfactory arrangements were made and my appointment was made, to leave Salt Lake City on the 26th of June.
The ?Southwestern States Mission? series uses the diaries of
six seven eight (as of 2012 Jun 17) missionaries who served in eastern Texas or, in the case of the Sister missionaries, in Kansas and Missouri, around 1900 to illustrate aspects of Mormon material culture, lived religion, and social History. The missionaries are Mission President Duffin, Elders Brooks, Clark, Folkman, Forsha, and Jones, and Sisters Carling and Cluff. The series is inspired by Ardis Parshall?s serial posting of the missionary diary of Willard Larson Jones at Keepapitchinin. Previous installment here.
 The following two entries were made while Duffin was in Utah. At the time there were hundreds of quorums of Seventy. I don?t know how much Duffin?s role as mission president affected the activities of the Ninth Quorum or the destinations of the mission calls: ?Sunday 13th our council of the 9th quorum of Seventies met and decided to recommend Prest. Freeborn D. Gifford and Oscar De Mill for a mission. Prest. Chas. A. Workman has already received a letter notifying him that he was wanted for a mission to the Southwestern States.? (Duffin, 1902 Apr 21 Mon); ?Sun 2nd Called the Presidents of the 9th quorum of Seventies together, and talked over with them the interests of the quorum, and decided to recommend Thomas Reeve and Joseph R. Naegle for Missions, and the presidents to visit Allen J. Stout and Nephi Workman and learn their feelings relative to taking the mission for which they had been recommended at a previous meeting of the council.? (Duffin, 1903 Aug 10 Mon).
 Hat tip to Andrea Radke-Moss for alerting me to the survival of some of Sister Carling?s documents at the Church History Library. (When I wrote ?hat tip? in the previous sentence I meant it like something from Animaniacs, with an extra dose of caffeine. Andrea?s showing me that letter is one of the top three ?document moments? of my life.) You might recall that I?ve written about Sister Carling before; look for future posts about Carling from both Andrea and me.
 The account is undated but occupies the first five pages of her mission diary. Sister Carling was living in Provo at the time. ?Bro. Brimhall? is probably George H Brimhall, acting President of Brigham Young Academy and father of Lucy Jane (?Jennie?) Brimhall Knight, one of the first two single sister missionaries; ?Bro. Reynolds? is George Reynolds, secretary to the First Presidency.