You'd need the catalog file and smart previews really.
You could have the Catalog folder in a Dropbox folder. The smart previews and previews would go to this folder automatically. You can edit on your computer and the edits would be saved to the catalog. Make sure that only one of you works on the files at a time.
This could be done by having the photographer export as catalog to dropbox and including Smart Previews also. Then when you've finished, they can use Import from Catalog to get your settings in.
Thank you! But would I need to export the whole catalog with original files as well as smart previews into my lightroom? Or how do I make sure that I'm only working with the smart previews?
You'd only need the photographer to select the current batch of files you need to work on and then use 'Export as Catalog' on them, with Build/Include Smart Previews checked.
Get them to do a trial run of a few photos (5 or 10) to test.