The only server functionality I'm aware of is syncing between devices. Zotero does offer hosting space for this, but you can host your own as well. It works with Webdav, so I have used my own hosted instance of Nextcloud for syncing.
Zotero is an offline software with syncing capability. They charge for extra storage beyond 200MB, which is their only streams of revenue, you can setup a WebDAV server and point Zotero there
For the attachments like pdf you can use webdav, but for the metadata you have to use their service [1]. They have the code repos online so it's open source, but no self-hosting instructions [2]. They claimed it to be due to technical difficulties [3], though imo they have no incentives to provide this given that they host a paid service themselves.
I've installed the app linked in TFA and it immediately asked me to sign in or sign up without any option to configure my own server address. But since it's still a beta version, the custom server/local only options are probably not implemented yet.