Cal.com works by everyone sharing their calendars to it, and it then suggests times when everyone should be available. It is great for doctors, dentists, teachers, companies meeting candidates, etc who need to set up meetings and want the process as automated as possible.
It can be self-hosted for complete privacy, or they also offer a hosted service which is free for individuals. Not only that, but it can connect to many types of calendaring services, and also has support for CalDav.
See https://cal.com/