It only does the recognition while the phone is charged and on the charger, so it won't really start until you plug in your phone overnight. As for cloud photos, i'm not sure, but it seems that it might work on the previews that are stored on your phone (and not wast time, bandwidth, and storage downloading everything from iCloud.)
Other than the fact that Apple said it is all local, i'm pretty positive it is local because it took a few days for it to actually go through my whole library. Google's cloud based search was set up as soon as everything was uploaded, so just a few hours later.
My only issue with it is that some sort of synchronization would be nice - having all my devices creating metadata seems like a waste, if they could sync via wifi on the local network that would be awesome.