[notoverthere]

This. If the link preview fails to load on the sender's device, it'll never appear for either party. The preview is sent along with the message – the receiving device never generates the preview.

If you're quick enough, you can see it in action:

Paste a link in a WhatsApp message. The preview might take about half a second to load. Hit 'send' before the preview has loaded, and it'll never show for that message.

Paste the same link again, but wait for the preview before hitting send. It'll stay attached to that message.