https://haveibeenpwned.com/Passwords
This is a separate feature from their email watching service, it also notifies people if their password was leaked, but you only send them a small part (called suffix) at the start of the hash of your password, and they return all the hashes from their database that also start with the same suffix, and then the code on client's side looks if any of the hashes matches the original hash.
By the way, haveibeenpwned accepts Bitcoin donations, which is a good example of Bitcoin's adoption and real use.
If you are using something very close to a random password, having one compromised should not affect your security on any other site and you can search by username to check if a database has been compromised