I can let someone else answer more fully, but I suspect the answer is "it's still experimental".
On the desktop you can get WxWidgets working for more traditional form based apps. For mobile I'm not sure what's available. I suspect Android would be easier to work with natively because of the Java target, but I'm not sure if anyone has done this yet. For iOS you can also do NME extensions to tie into native functionality, so I'm sure you can hook everything up - but I don't know of anyone doing this, so it might be a bit of work to get going. (Someone else feel free to correct me if I'm wrong, I have very little to do with native iOS or Android at the moment)
Another option could be to use HTML5 for the mobile, and package it up as a native app. (Either using PhoneGap, or there is a project that Cau
ê setup for embedding a WebView in NME: https://github.com/waneck/croxit). This is the approach I'm taking for an app at the moment, and I'm yet to see how it'll turn out. In the examples you mentioned, Facebook took this approach (HTML5 + JS) and did it poorly, and everyone complained it was slow and buggy - so they switched to native. Foursquare also took the same approach (HTML5 + JS), but did a much better job, and people tend to say the app is quite good - so you'd have to look into it and see if it's a good fit for you.
Hope this helps - I'd love for Haxe to tie into native controls really easily, but I don't think we're there quite yet :)
Jason