John N. Oswalt gives the following suggestions in his commentary:
The statement that the seraphim were calling to each other is probably an indication that the singing was antiphonal [i.e., ‘ alternate singing by two choirs or singers’1], but it may also be a way of saying that they were delighting with one another in the glory of God (Ps. 145:11).2
Antiphonal singing is recorded a number of times in the Bible: see, for example, Exodus 15:20-21; 1 Samuel 18:7; 21:11; 29:5. About 1 Samuel 18:7, James E. Smith writes:
The verse opens with these words: “The women who were enjoying themselves answered and said.” Apparently the women sang alternately, i.e., they formed themselves into two choirs, which sang alternate or responsive lines (cf. Exod 15:21).3
Perhaps something similar is happening between the seraphim in Isaiah 6:3.
Notes
1. ‘antiphonal singing‘ in Encyclopedia Britannica
2. John N. Oswalt on Isa. 6:3 in The Book of Isaiah: Chapters 1-39 (Grand Rapids: Eerdmans, 1986), electronic edition. Cf. Hans Wildberger on Isa. 6:3 in Isaiah 1-12: A Commentary (Minneapolis: Fortress Press, 1991).
3. James E. Smith on 1 Sam. 18:7 in 1 & 2 Samuel (Joplin: College Press, 1984), electronic edition