How did the Gift of the Holy Ghost operate before Christ came? In particular, I'm wondering how it operated among the ancient Jewish people, who only had the Aaronic priesthood, and therefore would not be able to give the gift of the holy ghost.
The Holy Ghost has many duties. Some are available to people before the ordinances of baptism and the Gift of the Holy Ghost (in other words: they are alwyas available), others are only available after both those ordinances are performed. Others still are unique (such as His involvement with the conception of Jesus).
His duties before the ordinances
The Gifts of the Spirit (healing, tongues, etc.). 1 Cor. 12:7
The Holy Ghost testifies...
- He brings things to rememberance and teaches. John 14:26
- See God before death. D&C 76:118
- To comfort us. Alma 17:10, Acts 9:31, Moroni 8:26
His duties after the ordinances
- The ministering of angels by the power of the Holy Ghost.1 2 Nephi 32:3
- Gives us authority to speak of Christ. D&C 63:61-64 (example: Acts 18:5-6)
- The Holy Spirit of Promise (Eph 4:30; D&C 124:124). It is by this duty that the Sealing power is made active and people are able to have their calling and election made sure. Before the covenant of Baptism there are no covenants to seal, it being the first personal covenant and therefore required to eventually have your calling and election made sure. Becoming "born again" (Moses 6:58-60; John 3:1-7) or receiving the "mighty change of heart" described by Alma is being sealed by the Holy Spirit of Promise. See JST 1 John 3:9; 1 John 5:1,4,18; Mosiah 27:25-28; Alma 5:11-16.
What is the gift (not the Gift) of the Holy Ghost?
Consider these words from the prophet Alma. He was talking to the people of Ammonihah about the nature of the Nephite people (before their descent into wickedness).
Having been visited by the Spirit of God; having conversed with angels, and having been spoken unto by the voice of the Lord; and having the spirit of prophecy, and the spirit of revelation, and also many gifts, the gift of speaking with tongues, and the gift of preaching, and the gift of the Holy Ghost, and the gift of translation; (Alma 9:21, emphasis mine)
Here the "gift of the Holy Ghost" is lumped in with the other gifts of the Spirit. That almost seems counter-intuitive until you realize that the power and manifestation of the Holy Ghost is something anyone can receive (remember that list of pre-ordinance duties). The Holy Ghost's influence on our lives is truly a gift, a gift that can be lost through iniquity, withdrawn to teach humility, or is sometimes simply withdrawn, perhaps to see how we will act of our own free will. An example of this last condition is the withdrawl of the Holy Ghost from Jesus Christ as He hung on the cross, because the choice to die a sacrifice for us all had to be His, and His alone.
This function, those duties, this gift, has existed from the days of Adam until now. And it was bestowed upon Adam before his baptism, and the official "Gift of the Holy Ghost". (Which means that the Gift of the Holy Ghost was available from Adam to his decendants until the general loss of the Melchizedek Priesthood during Moses' time. I'll get to that in a moment.)
And the Gift of the Holy Ghost is...
The "Gift of the Holy Ghost" is an ordinance, not a condition or state of being. The words of the ordinance, "receive the Holy Ghost" do not convey universal access to the Holy Ghost. Indeed, because the influence (or gift, as Alma put it) of the Holy Ghost can and will be withdrawn for many reasons (mostly iniquity), the ordinance cannot and does not create a special relationship with the Holy Ghost.
But it does mean two things can now happen that couldn't happen before: the ability to authoritatively testify of Christ and the sealing blessing of the Holy Spirit of Promise. It is these we are invited to receive by ordinance, this ultimate Gift through the power of the Holy Ghost.
TL;DR - So what does this mean for Mosaic Israel?
As a people, ancient Israel lost access to the blessings of the Melchizedek Priesthood. They had the authority to baptize, but generally did not have the authority to perform the ordinance of giving the Gift of the Holy Ghost. Moses had that authority (as all the OT prophets did, having the Melchizedek Priesthood) but general access to that privilege was withdrawn.
Which means Mosaic Israel had (and has) access to all the functions of the Holy Ghost save two: they may not yet testify with authority about Jesus Christ or be sealed by the Holy Spirit of Promise. Access to this gift through the ordinance of giving the Gift of the Holy Ghost must now be done vicariously as must the act of sealing their marriages, etc.
Please note that I felt this answer was plenty long, so I didn't go into detail as to when the blessings of the Melchizedek Priesthood were accessible but unavailable due to apostasy (such as just before the Flood), etc.
1 Moroni 7:25-32 teaches us about the duties and affiliations of ministering angels. I'm studying this now. One of the duties is to fulfil and to do the work of the covenants of the Father, which I currently believe would be outside the authority or duties of the Holy Ghost. Another is to prepare the way among the children of men, by declaring the word of Christ unto the chosen vessels of the Lord. That is within the authority or duties of the Holy Ghost. Verse 7 teaches they are subject unto Christ, but 2 Nephi 32:3 teaches they speak by the power of the Holy Ghost. My heart tells me there isn't a discrepency here despite not having yet tracked down all the details. I do know from scriptural examples that ministering angels often serve in roles that are traditionally given to the Holy Ghost. Thus, my current conclusion that before the ordinance only angels serving in the first duty mentioned (doing the work ... of the Father) may be seen by people (example: Cornelius, Acts 10) but after the ordinance the righteous may call on angels to minister to them through the power of the Holy Ghost (which is further supported by the Aaronic Priesthood Key to the Ministering of Angels, consider especially the Key to the Gospel of Repentance). It's a work in progress...*