Yes, everyone will need to install Tropy on their computer and need access to Dropbox with the project file and photos in it.
At the moment Tropy will always write to the project file when you open it (even if you make no changes) – this is something we’ll gradually improve in the coming months. In Tropy 1.7 the project file will only be modified when you actually make changes (this should make scheduling access much easier); we’ll also add support for a read-only mode along with additional improvements to syncing the project file.