Check links online is off until you turn it on (in setup or in Settings). After a message arrives, ask Google Safe Browsing whether its links are known to be dangerous. Only short anonymous fingerprints of the links are sent, never the links, your messages or phone numbers. Off by default.
Why turn it on: Google's lists catch new dangerous websites quickly, often before they reach the offline lists.
What it sends: only short anonymous fingerprints of links (the start of a scrambled code made from the address), never the message, the full address or your number. Google can't tell which link you received.
In the link preview, a link checked online shows "Checked online: no threats known. That doesn’t prove the site is safe." or "Reported as dangerous by the online link check"; without the check, "Not checked online".
If online checks fail for a while, the Security tab says so (see "Online link checks aren't working").