No, Ethos does not have a copy of your private keys. When you generate a wallet, there is a handshake with our server that sends a copy of the extended public key to us. As you may know, you can create seemingly infinite private and public key pairs when you hold the SmartKey. The extended public key stored by Ethos allows us to generate public addresses, track wallet balances, and support airdrops. This also allows us to tie your SmartKey to your Ethos account and prevent bad actors from using your SmartKey on another Ethos account.
*The extended public key does NOT allow Ethos control of your assets. You always maintain full custody of your assets.*