I only found out the proper way to announce Anastasia after watching synchronised swimming. I remember hearing that the Russian duet of Anastasia Davydova and Anastasia Ermakova always had their first names pronounced as Ana-sta-SI-a. Again, that sounds more Russian than the western pronunciation of Ana-stay-zhar or Ana-star-zyhar (the latter is more American).
Despite my British ways, I think is important to pronounce someone's name the way they themselves say it out of respect to them. It takes a while to get accustomed to the Russian stresses, but once you get it you won't forget it. It also helps when it comes to learning foreign languages. I am currently teaching myself Russian for when I go and it makes it easier to get your head around the phonetics.
T x