FusionAuth cloud vs self hosting?

Are the free and hosted cloud versions functionally equivalent?

Yes. FusionAuth Cloud starts with Community Edition as the base version.

You can purchase one of our paid editions for both the self-hosted or FusionAuth Cloud. Once you enter the license id for a paid edition, it activates the premium features in FusionAuth (including Breached Password Detection, LDAP support, Advanced Registration Forms).