Antony,
Thank you for reaching out at Apple Support Communities today. If I understand correctly you’re attempting to send a URL to someone via the Messages app on your Mac running macOS Mojave, and while it doesn’t appear for you in the preview pane, it appears for the recipient several times.
Have you tried using the Share option in the menu bar in Safari to send the links?
Share or post webpages in Safari on Mac
I am curious to know if the link appears for you when you use this method. If not, you might try some basic troubleshooting.
- Quit and reopen Messages
- Start a new conversation thread with the recipient
- Restart your Mac
Cheers.