Coinbase does not actually have your bitcoin.
They are just viewing the transactions associated with your wallet key from the blockchain record.
So it's never "off" or "on" Coinbase or the paper wallet.
Coinbase and the paper wallet simply hold the keys to access your wallet records and approve sends.
You can actually go and download an Electrum desktop wallet and import your private key into that and be able to access your wallet and do sends from multiple places/wallets. They all just update their history from the blockchain record and display it to you.