I used to sleep on my stomach, with my left arm under the pillow and head. In those days I would sleep like a baby. Then I got sick really bad one winter, and started having sinus problems. During this time I started sleeping on my back. Now I always sleep on my back. I sometimes flip over on my right side with my right hand under the pillow and head. Sleeping this way is very comfortable for me, but it makes my breathing more difficult at times because of my chronic sinusitis.
My tinnitus is mainly central. I do get fleetings from time to time, and when I do it is mostly during the day and waking hours, and almost always in my left ear which is also my bad ear. My left ear has a mild hearing loss, the right does not (or the audiogram lies). I also have middle ear muscle spasms in the left ear. First there is the TTTS which from what I understand mostly recruits the tensor tympani muscle. I now also have a secondary muscle spasm manifestation which seems to stem from the stapedius muscle and causes a distinct click sound. The stapedius muscle of my right ear seems to have been recruited in this loop now as well, although it is not as audiable and not as frequent.
Why do I tell you about middle ear muscles? Because, if I lay down on my stomach and left side of the head against the pillow, I can hear my so called "objective" tinnitus quite clearly. The TTTS would cause audible contractions at... maybe 70 oscillations per second, then a pause, then start off again. So you can see why I have another reason now not to lay with the left side of my head against the pillow. Laying on my back makes that muscle calm down it seems, or it is far less audible this way.
When I am laying on my back, I can still hear the stapedius muscle going off at random. But it also seems to be triggered by swallowing. This all adds on to my light sleep and prevents me from resting properly. But what can I do? Not breath and not swallow at night? Interesting enough, the stapedius muscle and the clicks seem to be mostly active at night. Perhaps because of the position I sleep in. The clicks are real, they are not subjective as the common tinnitus reference material or even research papers will have you believe. (You know how they often list the different tinnitus manifestations - 'including ringing, buzzing, voices, clicks, and even music'.) At least mine are real.
I have pondered this question of sleeping position myself, several times. I have not come to any definitive conclusion. There may be good reasons to sleep in one position or the other. It all depends on your own unique set of hearing disorders and other health conditions, like breathing problems as I have pointed out above. Try to experiment a little and see what works best for you.
For what it's worth, I should say that my tinnitus began as a low static noise that didn't bother me much at all. In fact I didn't even recognize this as being tinnitus. When it became tonal, it was the left ear that started blaring away first. Then it climbed up the auditory ladder and found a comfy spot in my brain and settled in. It has stayed that way since, regardless of which way I laid. But I do recall having some strange sensation in my left ear when laying on my back, and this was before I had the clicks. It sounded a bit like pouring water on hot rocks in a sauna. Or pouring whipped eggs into a pan. (Pick your favorite analogy.)
Just because you brought up this question, I might try sleeping on my right side tonight and see if the clicks will stop.