That said, I have no idea how a multi-node BBS would work, in terms of keeping state synchronized.
That's quite the assumption.
There were a lot of different BBS hosting programs. They wildly varied in what they supported and how they were implemented. Further even within a given piece of software the ways you could configure them and the consequences also varied. Even if a given software supported concurrent users on a single PC for various reasons a BBS might choose not to host that way.
Earlier: one PC per user, shared file system using a Novell network. Later: multitasking OS (Desqview, OS/2) or BBS software that natively supported multiple users (like MajorBBS.)
I ran a BBS on an Amiga for a while. The OS natively supported multitasking, but I only had one line. At least I could log in the same time as a user...