No unfortunately you are exactly right.
The collaboration features - nice as they are - only works in the browser.
That means if you're editing a document in Pages on Mac or iOS and choose to share it with others they have to use a browser to edit it, and you will be forced to switch to use a browser as well, if you want to edit the document at the same time.
It a shame!
The best we can do is to post a feedback request to let them know that this is a feature we want!
I've already done so myself.
It's a shame that I have to use the web version to collaborate with people. It kinda push me away of the standalone program, as I am afraid to overwrite changes (of which I get notified when someone changed the online version while I was working on the document).
But I guess it would be hard to keep track of all changes made if someone were working online and I was working off-line, without an internet connection. That would create a conflict that, IMHO, would be hard to solve.
But still, it would be nice to have the option of collaborating while connected to the internet if using the program, instead of having to use the web version.