For the chrome case, you can currently only access ~/Downloads. This involves changing our habits as /users/. If you need to access files from outside this directory, you need to open a terminal and copy/move files there yourself.
That said, for now.. the chromium package does actually have a external unveil policy in /etc/chromium/unveil.* for each process type (main, gpu, renderer, etc). But that won't necessarily always be there.