There currently isn’t a way to request a draft version of a page using the public API. We typically recommend wagtail-headless-preview, a mature and widely used third-party package.
Ultimately, it would be better if Wagtail provided a way to retrieve draft pages using some sort of one-time-use-token. This problem should become much easier to solve after autosave is implemented in Wagtail, since there won't be a need to worry about generating previews with unsaved content.
When autosave is released in Wagtail, generating previews will likely be less of an obstacle since the API would be serving up the latest changes in all circumstances. This is can be achieve using https://github.com/cfpb/wagtail-sharing/pull/47