There is no "official" position on this question, since the Baptist Faith and Practice is silent on the matter. And in a body as diverse as the Southern Baptist Convention, there is sure to be variability in practice.
That said, it's safe to say that "private communion" by a layperson is clearly outside the Southern Baptist norm for two reasons:
- The Lord's Supper should be observed in the context of the local church
- A minister should serve the Lord's Supper, not a layperson
Both points are rooted in the standard Baptist (and Protestant) viewpoint that the Lord's Supper is an ordinance for the church, not something given to individuals. Ray Van Neste writes:
To describe the Lord's Supper as a church ordinance is to assert that this rite was given to the church to practice and not simply to individual Christians. This is the understanding of the great majority of Christians across the history of the church. ("The Lord's Supper in the Context of the Local Church," 369)
In the Scriptures, Communion is part of the gathered worship. It is not merely a private act. [...] Thus Communion is not fitting in individual situations. (376)
Van Neste indicates that this was also the view of leading Southern Baptists like W. A. Criswell and George W. Truett. He quotes Criswell's Doctrine of the Church (105) as follows: "the pastor should not take the Lord's Supper to individuals and administer it personally."
Similarly, Bobby Jamieson writes:
Only a local church should celebrate the Lord's Supper, and they should celebrate it in a gathering of the whole church. [...] Despite the commendable compassion behind the practice, it shouldn't be "taken" to those who are homebound or in the hospital. (Understanding the Lord's Supper)
These authors assume that even in these "individual" situations, communion is being served by a minister. This is explicitly required in a number of Baptist writings, such as the London Baptist Confession (1689), still highly regarded by many Southern Baptists:
The Lord Jesus hath, in this ordinance, appointed his ministers to pray, and bless the elements of bread and wine, and thereby to set them apart from a common to a holy use, and to take and break the bread; to take the cup, and, they communicating also themselves, to give both to the communicants. (30-3)
Similarly, Millard Erickson writes:
At least some of the duly chosen leaders of the church should assist in the observance of the sacrament; the pastor should take the leading role. (Christian Theology, chapter 53)