How would they know they have to kill each other if they don't know about each other? Deciding wether to make yourself infamous could be a part of a strategy.
I did state "they only have a name to go by", I think that kind of implies that they are looking for one another, or using the name for something, unless they just like going about their daily business while thinking about eachothers names. I think you're just trying to nit pick on this one.
And they don't know the true physical appearance of one another, it would be a pretty boring fight if they instantly knew what eachother looked like. Plus remember, ALL of the inhabitance of the city know about Zero and Kira. If these two can't harvest information by speaking to people, then they've seriously changed since their last anime appearance.