What does "hide IP Address" on the new Monterey Safari do if you are not connected to iCloud?

That's "Private Relay", which is in fact a feature of iCloud+ and turned on from iCloud settings not Safari and is still in Beta at this time.

About iCloud Private Relay - Apple Support

The Hide My IP address while similar seems to be something else since it is handled directly within Safari Preferences rather than iCloud and appears to be available without iCloud+.

From what have read - with the Paid Subscription to iCloud + - the feature being discussed supposedly will Encrypt all Data from Safari via two separate connections . There am not sure if via Apple Servers or what.

https://www.macworld.com/article/348965/icloud-plus-private-relay-safari-vpn-ip-address-encryption-privacy.html

It does a "sort of vpn", by which you access a website as if you were in a different location, and there is no trace back to your original IP.

It does not seem to be doing anything if you are not using iCloud+

I checked and my public IP is the same regardless. So not entirely sure what its doing.