To meet the requirements of encrypted communication over private IP addresses between Compute Engine instances in different Google Cloud organizations, a Cloud VPN connection is appropriate:

Cloud VPN: Cloud VPN creates a secure, encrypted tunnel between your organization's VPC network and the third party's VPC network. This ensures that data transmitted over the network is encrypted and secure.

Private IP Communication: Cloud VPN allows communication over private IP addresses, which helps maintain security by keeping traffic within the Google Cloud network and not exposing it to the public internet.

Firewall Rules: VPC firewall rules can be configured to control the traffic that flows through the VPN, ensuring that only authorized traffic is allowed, further enhancing security.

By setting up a Cloud VPN connection, you can achieve secure, encrypted communication over private IP addresses between different Google Cloud organizations.

References

Cloud VPN Overview