It's many years since I had children but we were curious about home birth, and what I understood from looking into it was that there is, apparently, no significant difference in overall safety between home births and hospital births.
However, there was a significant difference in the reasons things went wrong. Because in a hospital you're more likely to receive unnecessary treatment due to the "cascade of intervention"; and interventions such as C-sections have inherent risk. While at home, you're more likely to have an actual emergency.