It should be shiny side out, particularly if the rest of it is foam or something with good conductive resistance (low thermal conductivity). By the time your temperature drops going through the insulation the surface temperature on the outside of the insulation won't be very high and you won't get a hell of a lot of heat loss from radiation, but anything you DO have will be reduced to very little when you have the shiny side out and every little bit helps.
If you have the other side out, it probably radiates heat a lot better than something shiny, so increases heat loss. And the only benefit of the insulation on the middle is the conductive resistance the shiny stuff offers, which probably isn't that flash (probably actually has high thermal conductivity).
But the best reason of all - never mind heat transfer fundamentals - is that it looks heaps blingier with the shiny side out.
Just some rough numbers, comparing my stainless mash tun (0.4m diameter, 0.6m high and ~1m2 of surface area) the heat loss from radiation from the bare metal at 50°C was about 140-150W. With 75L of water that means I would be losing at least 1.5°C per hour. At 65°C it was 250W, or about 3°C per hour.
With a product with an emissivity of 0.03 (i.e. shiny insulation stuff) that goes down to about 7W, or insignificant. The air/foam on the middle reduces the surface tempearture a bit (because heat doesn't get out as easily) and then the shiny surface poorly radiates it so it reduces heat loss in two ways and is theoretically 0.3°C per hour. Actual measurements are about 0.5°C per hour.
EDIT: Note that white is just as good as shiny, believe it or not
