Parent commenter never mentioned anything about comparing stored password hashes. What you do is block bad passwords at password set time by hashing the prospective password and comparing with HIBP. A prospective password you haven't accepted or stored or transmitted off the application server - common sense says that's not a privacy violation - and many giant companies including my employer do this routinely.
[Edit] Oh yea I remember HIBP has an online API. Don't use this. Take the HIBP dumps that they make freely available and compare locally. If not for reasons of privacy, for reasons of simplicity and removing an unnecessary external business/legal/software dependency.