If two black holes collide and then evaporate, do they leave behind two naked sigularities ore? If there are two, can we know how they interact?
Since classical gravity is much stronger than quantum gravity, the collision will take place on much faster time scales than the eventual Hawking radiation. So, the two black hole will experience a fairly violent process, and eventually settle down to a static situation which is most likely a new black hole. That black hole will evaporate on much longer time scales, eventually either evaporating completely, or leaving behind some long-lived "remnant". Nobody really knows what the black hole leaves behind after evaporating, but I don't think it will be a naked singularity.
The answer is neither two nor one. Black holes that collide and subsequently evaporate leave just a bunch of radiation. There is no remaining 'naked singularities'.
In the past there has been a lot of confusion about black hole evaporation. But questions like "what happens with the information describing the micro state of the black hole" have been answered. Key idea is the holographic principle and black hole complementarity. If you want to learn more about this, Lenny Susskind's book 'The Black Hole War' is a good starting point.
you can't produce any naked singularity in any process in 3+1 dimensions. This statement was originally coined by Roger Penrose under the slogan "Cosmic Censorship Conjecture" (CCC). A naked singularity would be a singularity that could affect the world around it because it would have no horizon; I will explain these words later.
Because dynamics seems problematic near a singularity - it's hard to calculate it by classical general relativity or even the descriptions of quantum gravity that we know - it looked like the Universe would be unpredictable if there could be "naked singularities", i.e. singularities that are unprotected by event horizons. That was viewed as a problem by Penrose (and others), a potential inconsistency that the Universe should avoid.
This thinking due to Penrose has been showed partly incorrect, at least in higher dimensions. A proper theory of quantum gravity simply can (and has to) produce predictions - at least in principle - what happens to an observer who can observe any singularity. And indeed, there have been counterexamples - thought experiments in higher dimensions - in which one can produce a naked singularity at the end.
As far as I know, it is still true that the CCC holds in 3+1 dimensions. One can't create naked singularities by any process - whether it involves 0 black holes, 1 black hole, or 2 black holes. Black holes usually have a singularity inside them but they are not naked exactly because they're black holes. A defining feature of a black hole is that they're holes that are black - which means that light can't escape from them. It can't escape from them because it is confined by the event horizon. The event horizon are the "clothes" in which the singularity is "dressed" - so it is not naked.
So despite some popular misconceptions, the defining feature of the black holes is the event horizon, and not a singularity. A singularity is just a typical addition that comes with the black hole "package". In Schwarzschild black holes, the singularity is spacelike. Nevertheless, once the black hole evaporates, nothing is left. At the very final stages of a black hole decay, the tiny black hole looks pretty much indistinguishable from a heavy particle species in ordinary particle physics (or string theory), and it decays to a few particles that we know from the colliders.
Two stabilized black holes, when they collide, emit some gravitational waves, and when they're sufficiently close (so that they don't just continue as two black holes), the rest of their mass=energy that hasn't been radiated away collapses into a single black hole that quickly gets stabilized as well. This black hole will proceed with the Hawking radiation and after a very long time, it disappears as well. Nothing is left. The research in the last 20 years or so has made it pretty much clear that there can be no remnants left etc.
All the best Lubos
The rules of general relativity give us a very specific set of rules for the interaction of the two black holes. We can integrate thee equations to know what would happen if, for example, we had two black holes orbiting each other, or on a head-on collision course.
The result that we get is that the two colliding black holes will not evaporate or form a naked singularity. Their horizons will coalesce into a single distorted horizon, which will then smooth itself out by emitting gravitational radiation. At late times, you will have a single black hole with a mass less than the combined masses of the two initial black holes (the difference in mass will be the energy that is radiated away as gravitational radiation). According to numerical simulations of collisions, no naked singularity should be formed for a generic collisions.
The answer is zero. When black holes evaporate due to Hawking radiation they will not leave any singularity. The singularities in black holes are inside the event horizon.
However, when two black holes collide their singularities do merge to form one singularity inside the event-horizon of the merged black hole. Perhaps that is what you really wanted to know.