How is RFC1918 address routing done between sites?

I've always wondered, how can you route traffic via private IP addresses between one site to another. For example, the first site uses subnet 172.16.1.0/24 and the second uses 172.16.2.0/24. Since traffic will go over the internet, so over public IP address space, how can the first subnet route to the second subnet?

Note that I'm talking about pure routing solutions if possible, using OSPF or BGP, so without VPN functionality. Or is a VPN connection always required in such cases?