Connecting to monogdb with string
As mention your error like 


  MongoError: failed to connect to server [private-network-ip:27017]


This error is what you would see if the mongo shell was not able to talk to the mongod server.

As per MongoDB documentation Connection String URI Format The URI format for defining connections between applications and MongoDB instances in the official MongoDB drivers.

The following is the standard URI connection scheme:

mongodb://[username:password@]host1[:port1][,host2[:port2],...[,hostN[:portN]]][/[database][?options]]


For further your ref here