I've been playing Judge Eyes since last Friday. Despite starting from chapter 2 (carried over my progress) and playing a bunch I didn't make much story progress over the weekend.
I think I just started chapter 6 (or 7) before going to bed yesterday, but on the other hand I finished all sidequests I unlocked over the 20+ hours I spent with the game and unlocked 30 friends.
So far my impressions are a bit mixed. On one hand, the story has some interesting moments and I'm enjoying it, but I'm not a huge fan of Kimtaku's voice/face acting (he looks and sounds bored 90% of the time) and the amount of running back and forth for both main and side missions. I guess the latter is to be expected from a Yakuza game.
As for the new systems, the tailing missions get REALLY old very quickly. Chases are a bit more fun, but once you've seen one, you've seen them all (at least so far). To make both less frustrating, I definitely recommend starting by unlocking SP skills making them easier, especially since they're quite cheap.
I'm also not a huge fan of the battle system in this one, compared to 0, but it definitely became a bit better once I unlocked the combo speed+ perk.
All in all, some quirky side missions aside, the full version is just more of the stuff in the demo.
If you enjoyed that, go for it, but if you were not a fan of the systems in the trial version, maybe think twice about buying it.