A hint may come from the Hellenistic-Jewish document Joseph and Aseneth [Collins, 1 Cor. commentary, 612] in which Joseph refuses to kiss the priestess Aseneth in greeting while she is still a pagan, but once she becomes a Jew he does so. If this is any model, then the destruction of social distinctions ("neither male nor female"), and the purity rendered onto each believer by fellowship with Christ, suggests that the greeting was exchanged between all.

A reader has added some notes of interest:

Just thought you might find it interesting that even today, in many non-western cultures that do not share "our obsession with the sexual", greeting one another with a kiss is still a common practice. So for example, throughout Russia, Greece, Romania, and the Middle East, it is quite common to see Orthodox Christians greet each other with a kiss - not a big smacker on the lips, mind you, but a light kiss on each cheek. As you say, it's merely the equivalent of a handshake - nothing sexual about it.

Likewise, throughout many regions of Africa, it is very common for people of the same sex (both Christians and non-Christians) to hold hands. Again, nothing sexual about it, just a way of showing friendship. It's really a shame that people in this country are so obsessed with sex that they assume that every other culture in the world (and throughout history) is as well.

I also wanted to mention, regarding the perspective "that this was a greeting performed in coordination with the liturgical rites of the church (i.e., the Lord's Supper)", that, at least in the Orthodox Church, this is still a part of the Liturgy. In the ancient Church, the "Kiss of Peace" took place among all the faithful of the congregation. Again, it consisted of a light kiss on each cheek. In the Orthodox Church today, the Kiss of Peace only occurs between the clergy during the Liturgy in 99% of the churches, although there are a few that still practice it throughout the entire congregation.