Yes, it entirely runs on your hardware. On upcoming NostrNet PWA, you should even be able to turn off your internet, and it will still work. There are numerous options, including 1 billion models that can also run on your phone. Click on ""model name" on top centre, it will show you the list of all the models. 

 thanks. on my mobile device it defaulted to one model. still, it lags quite a bit. haha