Apple Photos face recognition: cloud or local?
Is face recognition performed in the cloud or locally on each device?


According to Apple: https://support.apple.com/en-us/HT207368

“When you search your photos, all of the face recognition and scene and object detection are done completely on your device”

Now, about your issue, here are some things you should know and can try that might answer your question. 


in order for your devices to build up the curation of faces it needs to be plugged in (according to the initial pop-up you receive after purchasing a new device and opening the photos app)
As you take photos the curation process will always be running, since you have a huge database of images, allow your device to sit with the photos app front and center. 
As far as I am aware, Apple does not automatically add people to the People album (probably to save processing power). You must find a photo of the person you would like to see filtered from your library and enter a name. This will add them to the People album and begin sorting all images of that person immediately from all photos. 


I just tried this with my iPad and iPhone, and both devices have empty People albums until I add my name to my face manually and then all photos where I appear are sorted. 

Edit: Faces that appear constantly throughout your library might appear in the People album on their own, but you should still add whomever you desire manually for best results.