[@Burning Kitty]people would know that they were good friends and that they worked together a lot, so it isn't that hard to figure out.  I think she could do it.  
Obiwan knew who her father was when he saw how close they were after Padme got pregnant.

In one of the books Tarkin figured out that Vader was probably Skywalker, he just wasn't sure.  Vader calling for the Death Star to blow up a random backwater world was the last bit of evidence he needed to be sure.  He just needs to test it.