Multiplicity does not require a non private network. In fact it will not open the ports unless it is on a private network for security reasons.
How are you setting the network to private? If you mean you are using some sort of VPN to send all your internet traffic then you simply need to configure that to allow local addresses to bypass it.
If you are using a third party firewall then you simply need to ensure the ports 30564, 30565, 30567 are open on the machines.