What is the neurophysiological mechanism behind double hearing? A patient with sensorineural hearing loss can have the symptom of hearing "double" in the damaged ear. Not having a time-delayed echo, but hearing as if he (or other people) speak with "two voices" at once. 

Is there a theory about the mechanism behind this? Does it happen on the level of the malfunctioning hair cells, or because the brain has trouble remapping an incomplete input of frequencies, or something else?
Short answer
Damage to the inner ear can result in an asymmetric distortion in pitch perception between the two ears. This phenomenon can result in the same tone being perceived as a different pitch by the two ears.

Background
Damage to the inner ear (the cochlea) can lead to hearing loss. Hearing loss can sometimes lead to changes in perceived pitch. For people with unilateral cochlear damage, or asymmetrical hearing losses, the same tone presented alternately to the two ears may be perceived as having different pitches in the two ears (diplacusis). Sometimes different pitches are perceived even when the hearing loss is the same in the two ears (Moore, 1996). Diplacusis is associated with a higher perceived pitch in the relatively more impaired ear (Burns &amp; Turner, 1986). 

The precise origin of these pitch shifts is unclear, but are related to hearing loss (Moore, 1996). What is known, however, is that damage to the cochlea can lead to (Moore &amp; Carlyon, 2005):


Reduced frequency selectivity in the ear, i.e., a reduced pitch discrimination. This may alter the perception of pitch in complex tones; 
Altered velocity of the travelling wave on the basilar membrane. When the outer hair cells (the acoustic amplifiers in the cochlea) are damaged, the speed in which the acoustic wave travels through the cochlea may be altered, possibly leading to an altered perceived pitch;
Dead regions in the cochlea, where the inner hair cells (the mechanoreceptors of the inner ear) are absent. In this case a tone is perceived by adjacent regions in the cochlea. Because the cochlea is a frequency-to-place converter (basically a Fourier transformer), it may result in altered pitch perception; 
Reduced phase locking (Koppl, 1997) in the auditory nerve. Phase locking refers to auditory nerve fibers following the waveform of the tonal stimulus, which may provide pitch information (in humans this process is mainly confined to low frequencies, ~1 kHz and below). Reduction of this property may alter pitch perception.  


References
- Burns &amp; Turner, JASA (1986), 79(5): 1530-40
- Koppl, J Neurosci (1997), 17(9): 3312–21
- Moore, Ear Hear (1996); 17: 133-60
- Moore &amp; Carlyon, Springer Handbook of Auditory Research (2005); 24: 234-77