No. For instance, email correspondences are not considered public. Or non-public user account information like DOB, SSN, phone numbers, addresses, etc.
And, importantly, browsing history is not considered public. It's actually PII with the right machine learning.
"Handing to one other organisation in confidence" != "public"
The answer is potentially yes, if you're handling people's information then they have a right to (a) correct it and (b) delete it. You can't just maintain dossiers on people forever without their consent. However, it may be exempt if it's administratively necessary to keep.
It depends on how you're defining "public information." If you define it tautologically as 'information available to the public' then yes, because Google is crawling public datasets, and Facebook serves information to the public. If you define it legally, then it depends on the jurisdiction.
Public information isn't something established by nature, it's something created by statute.
And, importantly, browsing history is not considered public. It's actually PII with the right machine learning.