Process of buying property from a private individual - How exactly does it work?

We really did receive the addresses

Then don’t go broadcasting it, because it was not allowed. If purchase interest were already a legitimate interest and anyone could find out every owner with a phone call, one might as well just put the owner data online.

that someone wants to sell privately and gave us the phone number.

That is also questionable. Assuming the seller’s explicit consent, the caseworker is acting on behalf of the municipality and, even if no money changes hands, a quasi-activity as a broker is not permitted.
One can try it, but the employees of the authorities who release the data risk at least a warning. It just takes the right person to call or someone telling the right person.

I don't quite believe that the Hessische Verwaltung für Bodenmanagement und Geoinformation (HVBG) officially offers an ownership proof for the reason "Miet-, Pacht-, Kaufinteresse" via Geodaten Online if it is not legally "secured" in some way.

Both each for a fee of around 10 euros.

Perhaps that is sufficient along with the fact that the data of the requester is registered, to meet the requirements of the data protection law. At least these are two hurdles that prevent inquiries in the sense of "Who owns this property?" A phone call or a personal but anonymous visit to the office is certainly not enough.

Question neighbors. If you already walk into an authority office and ask "blindly," then you probably have no qualms about bothering the neighbors either :)

This has become common practice here. Friends who built in a new development last year have someone showing up about once a week asking if the house opposite, where the blinds are often down due to the owners being away for work, is for sale. Another friend got her house through such a doorbell campaign. A coworker writes to the addresses found in obituaries. Another coworker drops letters expressing interest in every mailbox where she likes the houses.