In Windows 8 the option can only be accessed via the charms bar. Go to networks there, you'll find an option that's something like "Turn sharing on or off". Despite the confusing and misleading name, that option is actually what you use to change between private and public networks.
Why they decided to remove this from the Network and Sharing Center and bury it even deeper, I don't know. Especially since it likes to default to private even on direct internet connections with a public IP address (so not terribly secure).