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I wonder whether these apps cache sensitive information like account numbers and balances. I think they shouldn't.

Yes they do - you don't need to plug in your device to see your current balance, even after power cycling the host device.

That being said - there's virtually no privacy in most cryptocurrencies. Your Bitcoin/Ethereum/Ripple/... balance is fully public.

The way they operate, I expect that there's HTTP logs of all activity. They do not operate in a way which is conductive to privacy.