Leaving aside all questions of context (this is likelier about personal interaction between believers and not about congregational singing) and the problems created by pulling a sentence fragment (one verse) out of a larger sentence and thought-unit, this verse does beg the question: What does it mean to speak to one another this way?
We've typically addressed this in congregational worship by singing a blend of songs that both worship God and provide mutual encouragement for the Christian way of life. The idea is that our singing together both worships God and instructs and encourages one another.
But how does this work functionally? For many of us, the "speaking to one another" during congregational singing means staring down into a book or speaking, at best, to the back of someone else's head in the row in front of you.
What if we took this idea literally? What if we took it as instructive for how to speak to one another (note that the verse does not use the word "worship")? What if it really is a more biblical way of fulfilling this teaching to rearrange auditoriums so we would be facing one another, so the "speaking to one another" could actually be done in a congregational setting?
Would be willing to do this? Are we willing to follow the meaning of this passage, or just derive a doctrine from it that justifies our practices?