The answer to this almost always is "that's totally up to you" but let me break down some of the pro's and con's my parents have noted to me.
In-home sessions are absolutely great for sentimental reasons, nothing is more personal a location than your own home. It's amazing for first time parents who are a little worried about the hooplah that is leaving the house in those first few weeks. But the cost is having to make sure the house is tidy and looking photo ready, some parents can't be bothered with the extra work of cleaning out a few rooms for photos (and that's totally understandable, you've got to try and sleep sometime!) Another thing to consider is that most homes aren't equipped with a lot of natural light, and not having that genuine light source can effect the overall ~vibe and tone~ of the photo.
And the opposite is true of sessions that happen in my studio. There's no lack of natural light with these 18 feet tall windows, so the results will always be pretty predictable in terms of how the photos will look. And you don't have to worry about cleaning up, you just have to show up and be! But that means having to leave the house and often times that can be a bit of a struggle with newborns!
There really is no right or wrong answer here, it's about talking with your partner and seeing which options seems most suitable for you and your situation!