So, I played a round as Unfallen with the intent of being peaceful. I became long term friends with nearly everyone and my diplomacy was taking off.  The AI started sending my messages that they thought we should exchange research (I wish they would just make a proposal rather than asking me to initiate it).  So, I go to the diplomacy panel, and notice... there are no technologies listed in the normal window.  I click on "demand" and can see the tech.  So, I demand some tech...

WAR.  The whole universe all gets mad at each other and everyone starts picking sides. The Family sends me a new Diplomacy message every 5 seconds...

The game hangs on the never ending pending turn. Game over.

Bug aside... how do I trade researched tech without needing to make a demand?

You need the "Perfect Trade" technology.   It's on the left side tech quadrant, towards the bottom in the 4th stage.  Required for trade agreements and technology trade. 

Research Perfect Negotioations.

Also demanding is damn weird.

Ah!  I see it now.  Thank you both.

Yeah, I have had very little luck with demands.  Usually, if I want something, I have to get them into a desperate situation, then ask for it during the ceasefire.