It depends on how much someone has mastered each keyboard, or their personal preference I think. I haven't seen any surveys or such about it. I am just speculating here, but people used to computer keyboards may go more for roma-ji input, while the phone-only texting maniacs may go more for the kana keyboard (see a video below of a kana keyboard power-user).
(The following is just extra stuff, about Japanese input on an iPhone, or Android phone for that matter.)
Roma-ji is pretty self explanatory. You just enter words phonetically on a QWERTY keyboard. As you type, suggested words come up in the top grey area and you can select those if they fit. This video shows it in action.
The kana keyboard looks like this - it's a 10-key keypad:
It looks a bit daunting, like you have to drill down through the stack of hiragana arranged by consonant sounds. You can do that by tapping on the keys repeatedly, or using the circular arrow key 2nd from the top on the left side. However, if you master the flick input method (which is pretty easy after a little practice; many people don't know about it though) it's actually the fastest way to write in all Japanese. The video here shows how the flick method works:
This person is very fast (sounds like a teenaged girl...what else)
Here's someone doing it with one hand using his thumb, which is what I do - or I use my index finger. (Warning: annoying clicky sounds)
However, entering the straight alphabet (necessary for URLs, etc) you need an English keyboard too.
I keep all 3 in my keyboard list (plus a couple others) but I use the kana one if I need to write out anything of some length in Japanese. Plus you have access to the bottom left key, which has a whole bunch of wondrous emoticons v(^o^)
Based on my personal observations, the majority of Japanese users use the 10-key keyboard. I imagine the reason is that they are already familiar with it from "dumb" phones.
Incidentally, it seems that very few people bother learning "flick" input, and instead press the keys repeatedly.
Again, this is only my anecdotal experience in the Tokyo area.