What solution would BEST facilitate private communication between the application team and the administrator's team?

Configuring IPSec is the best solution for facilitating private communication between the application team and the administrator's team. IPSec is specifically designed to secure Internet Protocol communications by authenticating and encrypting each IP packet within a communication session. This ensures that the data transmitted between the teams remains confidential and tamper-proof.

Using IPSec helps maintain the integrity and authenticity of the messages exchanged, making it suitable for sensitive communications between teams. It can work in both tunnel mode (for securing communication across networks) and transport mode (for securing in-host communications), allowing for flexibility depending on the specific communication needs.

The other options, while they provide some level of security, do not offer the same comprehensive encryption and authentication capabilities that IPSec provides. A host-based firewall may control traffic flow and prevent unauthorized access but does not specifically encrypt data. A RADIUS server is primarily used for authentication, authorization, and accounting rather than securing the communication channel itself. A Network Intrusion Detection System (NIDS) monitors traffic but does not actively secure or encrypt it, thereby leaving communications potentially vulnerable.

Thus, configuring IPSec stands out as the most effective means for achieving secure and private communications between the teams.