When I answered originally I didn't know if you mean Ditto transforming into Ash-Greninja or Greninja with Battle Bond transforming, since it was a little unclear. From the video it seems you mean the first.
A pokemon when it transforms is still a species identified as the original pokemon, meaning that all things unique to the transformed into species don't apply to the transformed pokemon. This includes form changes, exclusive Z moves, and (in this case) Water Shuriken's base power change and only hitting three times. For that video for example, it's still a Ditto, it just looks like an Ash-Greninja with the same stats as the opposing Ash-Greninja bar HP.
If that is the case, the Ditto exclusive items should still work after Transformation or Imposter trigger.
I will check again once I get my 3DS back by copying a Scarf Pokemon, but I am pretty sure it still stops working after Transform.
By the way, in Sun&Moon Ditto can spoil the Ability of the opponent in addition to his full moveset. You can do this by checking Ditto's stat changes window, as it will also let you check in the effect of your Ability, probably to help players that run Trace and have no clue of what "Magician" does or something.
It SHOULD be implemented that all enemy moves and ability get auto-revealed if Ditto switches in and you get the option to attack someone. In other words, if Ditto/Mew/Smeargle transform into a target and are not forced out before you can select another attack or opt to switch out, all enemy moves and Ability, will be revealed to the player, and to keep it consistent, they should show up as revealed on the screen. But if you get forced out immediately afterwards after transforming, you won't have had the option to see the moves and ability, and make them remain hidden. At the very least, you should get to see the ability and moves when checking on Ditto's stats when he is transformed.
Additionally, Ditto in gen 1, 2 and 4 should be able to access to any move learnable by Sketch, due to the Sketch and Rage glitches on Transform. After all, the Pomeg Glitch is allowed for, among other things, get Belly Drum level 5 Zigzaggon on Little Cup. Why not give Ditto some possible utility with that glitch?