At my hospital they have you go in at 35 weeks for all the paper work. I did not with my first and i had so much paper work when it was time for her to be born that I'm going at 35 weeks this time. Plus we have a brand new hospital and i have yet to even pull in the parking lot. I don't want to be in full on labor and not know where to go when i get there!
I filled out everything I needed beforehand through my doctors office. They just gave mea card to bring to the hospital when I went into labor.
I guess it depends on where your from. Im from Michigan. I was Induced and I signed one paper. I didn't have to register or fill out alot of paper work. I had completed my birth plan weeks in advance and just handed it to my midwife when I got settled into my room.
It is preferable to go to the hospital that you will delivering in and fill out all the paper work so they can have all your information and your insurance information. My cousin got the papers but she didn't turn them in until the day that she went into labor. To be 100% sure just call the your hospital L&D and they will let you know for sure.
Usually you want to register yourself in the hospital around 30+ weeks so they have everything on file and ready for you! If you don't have time and you're about to pop they will just take you in only needing your name and everything could be done at a different time