The Dropbox API allows you to interface with the normal Dropbox ecosystem, and when syncing with Dropbox there may be well more than 2 clients on a single account. To reconcile this, the Dropbox servers are considered the source of truth. So, when you're syncing with Dropbox, the intention is that you should get all changes from the server, as well as report all local changes to the server, as soon as possible.
Along those lines, while the Dropbox API doesn't currently offer a way to get a file ID or hash for a remote file (though we're tacking this as a feature request) it does offer ways to know if files have changed remotely so that you can update your local state. One way to use the "rev" property of a file at a particular path via the /metadata API call:
https://www.dropbox.com/developers/core/docs#metadata
If the rev has changed since you last checked it, the file at that path has changed.
Another way is to use /delta:
https://www.dropbox.com/developers/core/docs#delta
You can occasionally call this to find out about any changes you need to make to your local state to get back up to date.