I am of the camp that doesn't view Facebook as particularly private, or if you do your account is set up that way. I don't see any harm in send a Facebook message so long as you are aware that the odds of receiving a reply aren't high. Also keep in mind that even if you pay to have the message she can still control who she receives messages from. Even with the messaging app, I only receive messages from friends- the rest go to some inbox I don't ever check. If you can handle that and not try any other personal means of contacting her, then why not give it a try?