The phone doesn't have the memory/compute/cooling/or power budget to be a meaningful replacement for a desktop device which in turn doesn't hold a candle to the power or economy of doing it on the server side.

Herein privacy and utility are clearly in tension. The only sane thing to do is do it on the server but if you ARE going to do it locally doing it on a computer on the clients premises is the only way to actually do it because mobile devices power/compute/cooling/power budgets are so anemic whatever a synthetic benchmark says a device can do in the space of a moment or two.

I'm presuming that the ultimate strategy that a lot of players are going to actually follow is all server all the time wherein companies who are terribly concerned about privacy will rather than doing it locally do it on a server that they own.