Very interesting topic, premise, and discussions (both, answers and comments). I have thought about it myself, wondering about twins and if it would be possible to be identical in all aspects. What would it take to be identical?
Your premise attempts to eliminate all external factors, and all differences occurring naturally. That's what I wanted to do as well but the problem will still revolve around perception and identity. When you duplicate the person, as perfectly as you can imagine (same atoms, molecules, neural and quantum states, synchronization in time, etc), in that moment you are actually creating the most significant and fundamental difference in perception
You can try to eliminate all external factors ("infinite white room with no sense of position" - which is great btw!) and reduce all elements as much as you can imagine, you still create an unavoidable conflict the moment you place them "facing each other" because they immediately start perceiving each other - external factors with inherent differences ("their world" is no longer perfectly identical)
You may want to add new rules and conditions to your initial premise stating that they will move exactly the same, and exactly at the same nanosecond, symmetrically, but you cannot eliminate the most basic element - difference of perception and identity (looking at a real life mirror will still conjure distinct reactions and feelings)
The only way to enforce the idea to the point of perfection in identity is to reduce differences in perspective: in order to be driven and motivated to act exactly the same (as you imply when you attribute them with the same neural state and as I thought about it as well), you must make them PERCEIVE the same environment - the exact same external stimuli. That's only possible when they exist in the same physical space, and have the same sensory receptors: light, sound, touch, smell (all of them)
Only at that point you can achieve perfect identity, but you'll end up with the same person, having the same existential experience. If you're willing to accept this impossibility, there will be no need to communicate because they have the same thoughts. They cannot argue or fight with each other - it'd be like arguing with yourself. You can also argue with yourself, but it has to be sequentially (in time). You can have regrets, feel guilt, blame yourself for your actions but it's futile to hit yourself over the head - what's done is done (who's wins - your future or your past?)
Overall, it was a very good thought experiment: it compelled me to engage even though I don't feel relevant in this field with so many brilliant minds :)