I love this idea and I wish I would have done it when my daughter was that young. She's 12 now.
I think in the early years, until their teens, sharing it would be good. This way you can also turn it into a lesson. NOT pick it apart, but perhaps find a common word that needs to be spelled correctly? or simply capitalizing a name?
Once they get to those teenage years, I'm sure they will likely move away from a journal, but also when they do, it might be more private.
I've wrote in one from time to time since I was 20. I look back on it and laugh now, cry some, and generally feel good for writing down things, because now, my mom is gone and I can read back on those special times again.