What is the mechanism behind tinnitus?
Short answer 

The exact mechanism behind tinnitus (ringing in the ear) is unknown. 

 
Background  

Of the two theories you pose here, to the best of my knowledge the second one is the most widely accepted. It is a generally accepted phenomenon that whenever neural systems are being deprived of input, they start seeking new input, or even generate it themselves. For example, blind folks rarely have no visual perceptions at all. In contrast, they often perceive spontaneous visual sensations. For example, many blind folks report perceiving a persistent grayish background hue. In other words, it is never black in the eyes of a blind person. They also often see spontaneous spots of light. The most striking example, however, being Charles Bonnet syndrome, where the blind person may perceive complex scenes with faces and objects (Bhatia et al., 2013).

In the case of tinnitus, the perceived sounds are mostly not complex, but repetitive and simple in nature ('ringing', beeps etc.). It is thought that the deafferented auditory cortex actively starts generating sound perceptions. However, many other explanations have been brought forward, such as spontaneous activity in the hair cells in the cochlea and spontaneous activity in the auditory nerve fibers (Luxon, 1993). I will sum up a list of potential causes mentioned in Han et al. (2009):

Peripheral mechanisms

(1) Spontaneous otoacoustic emissions
Spontaneous otoacoustic emissions (SOAEs) are small acoustic signals generated by the electromotile activity of outer hair cells (OHCs) in the cochlea. While normally inaudible, SOAEs may be perceived as tinnitus. Tinnitus due to SOAEs is mild and is more common in subjects with functional hearing.

(2) Discordant theory
Tinnitus may be induced by the dysfunction of damaged OHCs while inner hair cells (IHCs) are still intact. For example, intense noise and ototoxic agents initially damage OHCs in the basal turn of the cochlea, and subsequently, if continued or repeated, affect IHCs, because IHCs are generally more resistant to such damage. IHCs are the receptor cells for sound transduction. In contrast, OHCs work as mechanical amplifiers, enhancing weak sounds by providing up to 50 dB, which can be evaluated by measuring OAEs. When OHCs die, cells in the dorsal cochlear nuclei (DCNs) show increased spontaneous activity due to the fact that IHC input is relatively normal, but OHCs input is decreased due to damage to the OHCs. This spontaneous activity is perceived as tinnitus. 

Spontaneous activity in neurons can occur through a host of mechanisms. Basically, a depolarization beyond a certain thresholds sets the neuron off to fire. In case of the disinhbited DCN neurons, the input from the OHCs is taken away. OHC input to the DCN is inhibitory,m as it is believed OHCs activate inhibitory interneurons. The exact mechanism is unknown, but a central inhibitory neurotransmitter is GABA. GABA can activate GABA-A receptors, that open up a Cl- channel, which in turn leads to Cl- influx into DCN neurons and hyperpolarization. When this input is taken away, you can imagine that DCN neurons depolarize and when this depolarization reaches threshold, the DCN neuron may increase firing Kaltenbach et al. J Neurophys 2002;88:699-714. Hence, as you question whether, in the absence of neurotransmitters, cells can fire action potentials, I say: **yes*. In fact, many neurons have spontaneous spiking activity, often with very regular frequency. These oscillating behaviors are often mediated by cation channels that gradually depolarize the cell up until its spiking threshold.     

(3) Crosstalk theory
Nerve compression (e.g., due to a tumor) may cause artificial synapses to be formed between nerve fibers of the cochlear and vestibular nerve  (crosstalk). The breakdown of the myelin insulation of the nerve fibers may further enhance coupling. Hence, spontaneous activity in the auditory nerve may be generated through vestibular activity.

Central mechanisms

Some patients may still perceive tinnitus after the auditory nerve is transected. Hence, tinnitus in this subgroup must be caused by mechanisms located more centrally from the auditory nerve. 

(4) Auditory plasticity theory
Damage to the cochlea enhances neural activity in the central auditory pathway, specifically in the temporal lobe in the auditory association cortex and inferior colliculus. This theory was touched upon further above in the first paragraph. Basically, when neural tissue is deafferented, it tends to look for new input or even generate activity de novo.

(5) Somatosensory system
The activity of the DCN is also influenced by stimulation of nonauditory structures and the somatosensory system has been implicated in development of tinnitus. Somatic tinnitus can develop as somtaosensory input may disinhibit the DCN.

(6) Limbic and autonomic nervous systems
The aforementioned theories cannot explain why some people suffer from tinnitus while others do not. More than 80% of those perceiving tinnitus for the first time do not associate the sound with any negative meaning and experience spontaneous habituation. However, if the first perception of tinnitus induces high levels of annoyance or anxiety by association with unpleasant stimuli or with periods of stress and anxiety, tinnitus might develop into a problem. 


References
- Bhatia et al. Delhi Psych J 2012;15:423-4
- Han et al. J Clin Neurol 2009;5:11-9
- Kaltenbach et al. J Neurophys 2002;88:699-714
- Luxon, BMJ 1993; 306:1490-1

Further reading
- Action potentials in the absence of neurotransmitters: how-does-optogenetics-work