Very minimum, though. Imagine the equivalent of FeatherWallet to mobile version, just as Electrum for mobile: https://bitcoin.org/en/wallets/mobile/android/electrum/. In Feather, you do not reveal to the public nodes the recipients, nor the amounts transferred. You neither reveal your inputs, although there's a chance the node executes a very rare (and costly I think) attack which can reveal the true input of a transaction, but that's detectable by the client.

The problem with mobile client would be bandwidth and verification time. You would need to download the entire blockchain since your wallet's block height. That's feasible to do for a modern computer, but a smartphone has a slower Internet speed and is less computationally capable to verify all those blocks.