The built-in modules in nodejs have some support for debug flags. You set them in the environment variable NODE_DEBUG as in NODE_DEBUG=http and then those modules will output a bunch of information about what they are doing to the debug console.
Some that might be relevant here are:
NODE_DEBUG=http
NODE_DEBUG=net
or both together:
NODE_DEBUG=http,net
I've found that these will not give me the same level of detail about the data being sent and received as the network tab in the browser, but they will give you the general idea of what requests are being made and to whom.
Depending upon your operating system, there are numerous tools (often called packet analyzers) that let you monitor network traffic on a computer across all processes with varying smarts about how much they can parse the protocol to give you a higher level view of what's happening. A couple tools like that are ethereal and wireshark.
While you can ultimately see anything in the network traffic with these tools, my experience is that its often a bunch of learning with them to be able to zoom in on exactly what you're looking for.