So how does this sync exactly? Can I use an iCloud folder, an iOS filesystem provider, or do I need yet another service to have my notes on my desktop as well?
File Provider is something we'll attempt to implement sometimes in the future to support other sync providers.
More info available at https://help.obsidian.md/Obsidian/iOS+app#Sync
Apart from that they also offer a sync service. And for the technically minded there is also the obsidian git plug in.
I guess the options will stay limited, as the author wants to sell his commercial sync service.
Just putting a note in here to say that that's unlikely. On Android, you can use a lot of different approaches and I actually didn't need to use Obsidian's sync service. On iOS it's more complicated than that because of iOS's way of handling files. Building the integration of any other sync providers into Obsidian itself feels like a giant time sink for 2 people.
Basically, the options will stay limited but probably for actual feasibility challenges rather than personal gain.
(Also, sidenote: assuming gender is maybe not necessary: his => their)
Joplin, as an example, allows syncing via WebDAV. Which you can easily self host.
I know that Joplin works differently. Obsidian also has integrated sync capabilities, but it’s a proprietary protocol and you can’t self host it. And I guess this is a design decision of the authors. To sell their service. They probably need some money to pay their bills.
On a personal note, please don't do this. I understand people choosing their preferred pronouns, but don't force it on others unless they have stated preferred pronouns too.
Quick edit: went and looked at the developer team (https://obsidian.md/about). It's actually a man, a woman and two cats, so "their" is clearly appropriate.