When you’re planning on sending some coins and tokens — yes, your Ledger Nano S or Ledger Nano X needs to be connected to your computer or mobile phone(through Bluetooth with the Ledger Nano X) with an internet connection to be able to send out funds. You need an internet connection when sending funds using any wallet — may it be a mobile wallet, a desktop wallet, a hardware wallet, or an exchange wallet.
No. Your Ledger Nano S/X does not need to be connected or to be online for you to receive funds. As long as you know your wallet’s address, you can freely receive funds on that wallet at any time.
You do not lose your funds if your wallet is offline. This is quite a common misconception with beginners in the cryptocurrency space, as your wallet can even generate new addresses even when offline.