No you dont store your private key!

There is two key on nostr

Npub & nsec.

Npub is a public address. When you register your npub on getalby, you kinda get a username

See mine : ihsanmd@getalby.com

When you activate the LN wallet on getalby.com

You get a custodian LN wallet with ihsanmd@getalby.com as the LN address

You should never gives your nsec (private key) at all cost to anyone.