As you likely already know, in Austria in the case of a normal vaginal birth, it is the Midwife who delivers your baby, rather than the doctor.
As such, it helps to have a good midwife, with whom you have a good rapport, and who understands your birthing wishes ahead of time. All of the hospitals have a team of qualified midwives on hand, so if you decide not to go the private route, you should still be in good hands.
However, if you want to establish a relationship with your midwife before the big date, it may make sense to hire a private midwife. Some of the best sources for these are listed here.