If you are looking at the frame of reference of the flashlights, then the answer is indeed 2$c$. How can this be so, when nothing travels faster than the speed of light? Ah, but 2$c$ isn't a rate at which anything is actually travelling.
You asked for the rate at which the distance between the photons of the North Beam (the leading ones, say, and supposing we could know their position to some fixed precision; asymptotically speaking, any fixed precision will do) and the photons of the South Beam (ditto) is increasing, in the frame of reference of the flashlights. Nothing is travelling at 2$c$, relative to the flashlights, if special relativity is correct; both beams are travelling at $c$ relative to them.
In fact, the two beams of light can accumulate distance between them at any rate greater than $c$ up to an upper bound of $2c$ for different moving observers, depending on the (sub-light) velocity they have. If their motion is parallel to the line of transport of the two beams, they will also "observe" (they aren't actually observing the light, after it has departed) a distance-accumulation rate of 2$c$. If they travel with some lateral velocity to the light-beam-axis, the rate at which the two beams separate will be less than 2$c$, but strictly greater than $c$.
Again, these rates aren't rates at which anything is actually travelling; these are rates at which a notional line-segment, with end-points defined by two moving objects, is increasing. In Gallilean relativity, that rate of increase of the line-segment happens to be the same as the relative speed of the two endpoints relative to each other; and the rate at which that line-segment grows will be identical for all inertial observers. However, while we can still talk about the rate of growth of that line segment in special relativity, it neither grows at the same rate for all observers, nor does the rate observed by any one observer necessarily correspond to the relative speed of the end-points away from each other relative to the end-points — which is the only way to turn this rate-of-increase into a rate-of-travel.
Note that I mentioned that the rate of growth of the line-segment can achieve any rate of growth between $c$ and $2c$. We can actually find a frame of reference (albeit a somewhat degenerate one) in which the rate of growth between the two beams is precisely $c$; one example, of course, is the frame of reference of the North or South beams themselves (and for which these speeds finally does represent a rate of travel of the beams relative to each other).