Can someone please explain how is the dual license work? Should there be any differentiation in the product, for example, community edition vs enterprise edition?
Also, how is the sponsorship work? For example, OBS Studio has Facebook and Twitch as sponsors.
But it can also be by use case. MongoDB's has extra requirements for those who run services (database-as-a-service) and thus who compete against MongoDB, the company. https://en.wikipedia.org/wiki/MongoDB#Licensing
Sponsorship is hoping companies rely on your software enough to open their money pockets. Or to have a strategic interest, e.g. by keeping something open/free that a competitor charges money for. https://letsencrypt.org/sponsors/
Will people pay for a license if there's no differentiation? (In HN we read a lot of stories where even donate button is not working well)
Also, will people pay for a commercial add-on when the product itself is free? (open core model)
That is really a separate issue. If there is no differentiation from its competitors, then it may not be used at all.
People aren't generally the customers of dual-licensed software; usually companies that ship proprietary software are the customers. The typical model for dual-licensing is to allow the code to be used in an open source project for free but charge for the code to be used in a proprietary project.
Support: RedHat
Dual License: Qt/Nokia, BerkeleyDB/Oracle, MongoDB
Trojans/Spyware: Android, Chromium