I used private for my whole pregnancy, but at the same time, I also registered at the public hospital. I went to public for a few times and then stopped. I did so because it was recommended by a friend who had complication before, and at that time she only went to private and thus she had no record at the public. When she was brought to the operating theatre which was an emergency case, she was asked to provide all her perosnal data.....and it took ages for the staff to complete the registration form!!
Even though if you don't plan to use public, it is no harm to get registered with them because you'll never know what happens (touch wood!)! Anyway, congratulations and all the best to you and yor baby!