Excellent feedback in this thread!
If I may, I'd like to offer my own observations: first, I believe that modal music does
exhibit tonality, and that it often does so via a 'functional harmony', albeit one that is particular to the mode being used. In my experience, this can apply equally to Impressionist - and some early 20th-century - passages as well as to the harmonization of modal melodies, as found in the popular and folk music traditions of Europe and the Americas -- musical traditions in which the patterns of modal functional harmony are most easily observed.
As a brief example, let's look at the dorian mode, which is widely used in what is often (some would say 'inaccurately') called 'Celtic' music, as well as in the folk and popular musical traditions of most of Europe. The chord qualities in dorian, per scale degree, are:
i (minor), ii (minor), III (major), IV (major), v (minor), vi0 (diminished), and VII (major)
As has previously been explained in this thread, the absence of a 'leading tone', coupled with this quite different order of chord qualities, can make for some very ineffective chord progressions and cadences -- IF one insists on using those progressions and cadences that are particular to the major-minor system that formed the basis of common-practice classical music. Rather, musicians and composers use progressions and cadences that are specific to dorian.
Since the only difference between dorian and the (natural) minor scale is in their submediant (sixth) degrees -- the dorian's is 'raised', relative to that of the natural minor -- it is natural that one finds at least some notable presence of that degree in both dorian melody and accompaniment. While dorian's ii, IV and vi0 all contain it, it is the IV that is most most commonly used to convey the 'dorian sound' via that mode's submediant degree.
Since dorian lacks a leading tone, it's v (minor) cannot parallel the use of V (major) that is so crucial to the functional harmony of the major-minor system. Accordingly, the major-minor system's 'authentic cadence' (V - I) cannot be replicated in dorian; instead, dorian music tends to use VII - i and/or v - i. Neither progression is as powerful as the major-minor authentic cadence, but they are effective enough in conveying formal design, and - of course - are far more effective in conveying 'dorian'! Both progressions are used as 'closing' cadences in the traditional tune "Scarborough Fair". Dorian's equivalent of a half cadence
ends on either v (minor) or VII; one can find instances of both in "Scarborough Fair" -- eg, III - i - IV - v, and i - III - (III-VII-III-) VII.
Since seven different modes are available for any given key signature, it is essential that tonality be firmly established, and easily perceived -- otherwise, any given passage can seem to belong to any -- or none -- of the available modes. For this, the same basic techniques are used in modal music as in major-minor music: the tonic degree must figure prominently in melody, as should other mode- and tonic-defining degrees (eg, the dominant in major-minor, the supertonic and subtonic in phrygian, etc, etc); consecutive, step-wise motion (in the same direction) should be used to create melodic direction - direction that should be used to enhance the sense of tonality; harmony, that is appropriate to the mode, should be used in ways that support these melodic characteristics; and so on. . .
Because each mode has its own, specific structure, each also possesses its particular melodic and harmonic features, and what works in one mode may not work well in another. The solution: try to find - and then study - relatively straight-forward examples of music for each mode; dorian, aeolian, and mixolydian are ubiquitous in western European (and, by extension, N. American) folk traditions, while phrygian -- and maybe
some lydian --can be found in some Mediterranean cultures. Locrian is another story!
As eloquently stated in this thread, Impressionist and early twentieth-century composers went way beyond this basic type of harmonic treatment of modes, using, among others, techniques sometimes described as "organum" (eg, Debussy), parallel chords (that quickly transcend key), and all types of chromatic inflections that often make modal identification difficult. However, I believe that one can find, even there, deliberately-established tonality, and functional harmony; from one perspective, one could argue that these composers brought to modal passages the same sophisticated compositional techniques they routinely applied to the major-minor system.
For someone -- say, a songwriter -- who wishes to experiment with modes, I would recommend first analysing some of the hundreds of modal folk and popular tunes that are so readily available -- look at their melodic constructions, their chord progressions and cadences. These are, for the most part, simple constructions, and relatively easy to analyse. Once you have these basic models firmly established, take a look at what the Impressionist composers (and some of their predecessors) did with them; as has been pointed out in this thread, mode use, in those compositions, will be less obvious, and often quite transient (eg, Ravel's Ma Mere L'oye, fifth movement, begins in C major, moves to A dorian, back to C major, then into E phrygian - with one chromatic inflection in the melody, - then into C# phrygian, G# phrygian, etc.) Of course, it'll be very helpful if, prior to this, you also have a good grounding in common-practice harmony!
Finally, it should be noted that jazz
musicians use modes in an entirely different way: for them, modes are not
the basis for composition, but notes to be used against any particular chord
-- in short, a kind of memory aid for the intense, and highly sophisticated, melodic and rhythmic work that goes into jazz improvisation.
I hope this is of some use.
All the best,