Definition: Private Peering is peering across a dedicated layer 2 circuit between exactly two parties, typically using a fiber cross-connect or a VLAN between two parties at an IXP.
There is typically a nominal cost to Private Peering (a few hundred dollars per month for a fiber cross-connect in the U.S., for example), where circuits are typically more expensive (a few thousand dollars per month for a 10G circuit, for example).