When you sign up, Legend creates a self-custody wallet for you with Privy. Your funds are in that wallet on Hyperliquid. Legend cannot move your funds without your approval.

You sign in with Google, X, Telegram, email, a passkey, or an existing wallet. That sign-in method gives you access to your wallet.
You do not have a seed phrase to write down. If you want a backup, you can export your private key.
Your wallet has one address. That address is the same on Hyperliquid and on every EVM network.
Trades settle onchain on Hyperliquid, so anyone can verify your balance and your trades on a Hyperliquid explorer.