Are they sharing search terms, type of doctors selected, symptom checking info which is common on these websites? Especially if all data can be use to uniquely identify this is pretty problematic.
If you've used google analytics, you know that a lot of this would be implicitly shared. like the top comment, I think hospitals just aren't thinking about how disastrous this is.
Facebook widgets definitely attempt to track this sort of information. Anything that goes in the URL bar is trivial to grab, but pretty much anything typed on a page is also snaggable.