You use Dropbox's already existing functionality to make a copy file. You then compare the changes and if they don't conflict you merge the files (while backing up both just in case of error). If they do conflict you send a message to the tray application which displays a popup showing the conflicts and asking for a choice. This isn't rocket science.
There are certainly places where that wouldn't be feasible like video editing. But I don't buy the argument that "because you can't do everything you should do nothing". Again I point out this took me a weekend to do with office documents and I didn't work through the night to accomplish it. So you can't really tell me it's out of the grasp of a company that has millions in funding.