Folks writing about stringed instruments will often use the terms “bass side” and “treble side” to uniquely identify the two sides of an instrument. You'll occasionally see literature which asserts or implies that the bass notes and treble notes each originate from their respective sides of the instrument. Enough is known about the sound producing mechanisms of instruments to demonstrate that this is false. This has some implications for instrument design and begs the question of why some instruments designed around this erroneous notion seem to work so well.
Last updated: January 21, 2019
OK, the first thing to do to get an understanding of why the bass notes and the treble notes do not radiate from their respective sides of an instrument is to see how the plates of an instrument and the air inside actually vibrate. I'd highly recommend the animations of the basic modes of a guitar that Dr. Daniel A. Russell of Penn State has put together here. Take your time looking at these, and be sure to run the animations. Notice that each of these modes of vibration were induced in the guitar when it was excited by a specific frequency. Notice that none of these modes of vibration involve just one side of the guitar or the other. Another thing to note is the short descriptions that describe how much each mode of vibration contributes to the overall sound of the instrument. Modes that move a lot of air contribute heavily, while modes that do not, do not. Note also that the dipole modes, that is, the modes where one part of the plate moves up while the other moves down, are not particularly good radiators of sound, because the movement of the two parts tends to cancel each other out - when one side is pushing air out, the other side is pulling it in, and the net effect is that no air is moved.
Not noted on Dr. Russell's page is the fact that each of these modes radiates sound at a reasonably broad range of frequencies. These ranges overlap, so the instrument is capable of producing sound over a wide range of frequencies. There are peaks and valleys in the frequency response, but in a good instrument these are not severely high or low and so do not detract from the tone of the instrument. It is possible (even likely) that this unevenness in frequency response actually enhances perceived goodness of tone.
From the animation it should be clear that, since there are no modes of vibration in which one side of the instrument is still while the other side moves, no range of notes can come strictly from one side of the instrument. But in some instruments it is the case that some of the dipole modes of vibration are more biased to one side of the instrument or the other. Check out Mode #8 - the (1,0) mode (315 Hz). The text here says that the nodal line (the line that indicates a place where the top is not moving) tends to follow one of the diagonal braces of the top. As such this nodal line does not run right down the center of the instrument and radiation from this mode may be biased more to one side than the other. In this case this dipole mode may actually contribute more to the overall sound output of the instrument than if it was symmetrical. And if so it may be considered that the sound produced by this mode comes from the side with the greater displacement. Yeah, that's a lot of mays and maybes.
Note that the instrument used to derive the animations has an asymmetrically braced top. A number of instrument builders design tops with some asymmetry using asymmetrical bracing, asymmetrical brace carving, asymmetrical top thickness, or even tops composed of different materials. No matter the builder's intention, this asymmetry may result in more sound output from an otherwise non-contributing dipole mode. In summary, bass and treble notes are radiated from both sides of the instrument, but it is possible that in some instruments a small component of mid and high frequencies are radiated a bit more from one side or the other. This is not likely to be enough to support the supposition that there is a bass and a treble side of the instrument as far as sound radiation is concerned.
Now, to confuse matters even further. The supposition that there is a bass and a treble side of the instrument always seems to assume that the bass notes come from the side of the instrument with the bass strings and the treble notes from the side with the treble strings. But what happens if you string the instrument backwards (or some would say left-handed). If there was a side that was slightly biased towards the higher frequencies in an instrument that side will not change when the stringing is reversed. In fact if you do this, or if you string up such an instrument with, say, 6 identical B strings, you'll be hard pressed to be able to tell any difference at all in the tone of the instrument related to the location of the strings.