> Anything behind a login gate is private data for that registered user only

That's quite the claim, if only the login gate were either always there or indeed always not.

Presuambly such "private" data ought not to be being indexed by search engines and returned to users who search?

"site:instagram.com" is of the order of 228 million pages on google.com, and "site:facebook.com" is another 422 million.

Depends if another user can also access it, or whether the original author/owner of the data in question intends for it to be public. In Facebook's case, there are permission levels you can set on posts, including a "public" option (which isn't actually public though and will require a login anyway, but it can be any login) which would settle that debate quickly - hell I wouldn't be surprised if that option were to be hidden as to not acknowledge that a particular bit of data was explicitly posted for everyone to see.

but you make it public for everybody with the publicly accessible login so it wouldn't be considered private data for the same reason news outlets can use your instagram images and share it widely without your permission.

you can't throw up a login screen but then allow people to post themselves that ends up in public domain because the login does not distinguish from public or permissioned user authorized to view your selfie pics.