I recently had a discussion with new bitcoin user and he asked how to get started with bitcoin. I directed him to Bitcoin Core download page and explained about disk usage and bandwidth requirements. Disk requirements could be lowered with pruning but his monthly bandwidth limit does not allow to download blockchain in reasonable time and using bitcoin client for him does not seem to be an option. I was unhappy to do it but I directed him to some online wallet sites. He might not feel importance of bitcoin private keys and he would treat bitcoin as an digital cash without any difference from website that says 'you have won thousand bucks'. 'And where is cryptography?'

Use of bitcoin client would be easier for him if all data expensive operations are done by third party client (for example my client) and his client would be able to verify found transactions and and them to his wallet. Closest data to private keys that would be transmitted would be public address. Adding watch only address from his would work in same way. It would be less safe than usual full node but safer that online wallet website when you need to guess where your private keys are hanging around and with who. This would need additional both side identification with preshared public keys to prevent host abuse.