A computer realizes this distribution by giving every task attention in turn or let itself interrupt for tasks with a high priority. The nature of the task determines how long it is allowed to use the attention of the central controller. After that the thing that has to do the task is left alone for a certain period of time. The switching is done so fast it seems the tasks are controlled simultaneously.
The human body also has automatisms capable of going on their own for a certain period of time after they have got a control signal from the brain. This is the trick to control both hands or sing and accompany simultaneously.
For this purpose we have learned to restrict our number of chords and moreover, in a restricted number of positions and a restricted number of initial keys. This way we will have a possibility to put the left hand on an automatic pilot when we want to concentrate on the melody in right hand. For the melody we also have short melodies, like riffs, to facilitate the task of the right hand.
Another aspect comes to the aid of our problem, viz. it is not desirable to
make music too complex. During the changing of melody notes it may be pleasant
not to change chords also. This way the melody gets all the attention.
Click to hear what I mean.
The other way round, the changing of a chord sounds better if the melody note
stays for a moment.
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This aspect helps us to switch our attention fast between both hands, in
general between melody and accompaniment. We do it as fast as our other
controls (walking while talking, etc.).
Up to now I assumed the melody in the right hand and the chords in the left,
but these tasks may be divided in a different way among the hands. E.g. you
could take one note of the chord in the left hand and the rest of the chord
together with the melody in the right.
Click to hear what I mean in the song 'I can't
give you anything but love'.
If we have much automatism in both hands the combination may be complex.
Click to hear what I mean.
For some combinations division of attention is hardly necessary, e.g. if you
play note e together with chord C.
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By experience these combinations grow, as also automatisms, of course.
Rhythm may take advantage of the alternation of attention, as it is based on
time gaps. If we don't switch inaudibly fast, we may put these gaps in
the form of rhythm.
Click to hear what I mean.
A nice incidental is the pleasant tension you may have when the melody is
ahead of the chord to come.
Click to hear what I mean.
For a modulation the melody may even be ahead of the key to come, by
playing a note belonging to the new scale, but not to the old one.
Click to hear what I mean.
Also the melody may stay behind the accompaniment giving the feeling
the solution is postponed.
Click to hear what I mean.
Accompany your own singing has similar aspects. Here too it is an
advantage if the accompaniment is unpretending during the singing, in
order to give the voice full attention.