With the game being on Steam - it wants to stream.

When I play with friends, it's kind of pain to stream it. It'd be great to create a network game option where it's just a server.  Would that require a second steam account? Is that allowed? 

You can set it up as a server only, but it cannot be run headless. I know of at least one case where this was done via AWS, It was discovered there was no way to save/shutdown but I believe this was fixed - certainly not tested well but I think they were satisfied. 

EDIT: I would recommend no one be logged in when you shutdown/save. It will auto save as well now.

I am assuming you have a stand alone version from itch. It is not a license issue with you running a stand alone as a server then connecting to it yourself with another instance. But you have to run server/client as separate instances - both have to have unique reference directories (a second installation - not the same one if you are using one machine to do it all).