I think the best solution is a version-control system such as git with a hosting service like github. Github is free for everyone if your repository is public, while academic users can get free accounts that allow private repositories as well. A github project has an "issue tracker" than can be used like a collection of "chat rooms" to carry on multiple discussions at once, and you can also add comments and carry on discussions about particular lines of a source code commit or pull request.
(Using version control allows many people to work on the TeX code at the same time without losing history or overwriting each other's changes. A real-time collaboration site like overleaf also allows multiple people to work on the file at once, but limits you to using their in-browser editor instead of much better ones like AucTeX, and doesn't save previous commits in an explicitly labeled way or allow other useful features such as branching and forking. There's a bit of a learning curve, but I think most mathematicians should be able to figure it out.)
Another tool I've found useful for collaboration is the LaTeX package
fixme, which lets you all insert comments into the document itself (labeled by author, and even color-coded). Not so great for carrying on long conversations, but good for making notes about things that need to be fixed that will be visible in just the right place and not get lost.