Would it make two separate phases? or would it make something like and emulsion?
There is no "or" in the first place. An emulsion is a two-phased system, it is just that one phase comes in a form of small droplets. Small as they are, however, they are pretty big from molecular point of view.
Now to the point. If you stir the liquids vigorously, you'll have an emulsion. Short of that, they will attempt to minimize the surface, and the shape with minimal surface is a sphere. Indeed, that's what you will end up with. It's been done before. Look up the Plateau's oil drop experiment: they put oil in a water/ethanol mixture with ratio tuned so as to match the density of oil, and behold, it just floats there in a huge spherical drop.
If both liquids get in contact with the wall, the geometry might become more complicated. The phase separation surface will develop certain contact angle where it touches the wall, and the rest of it will form some kind of minimal surface, which can be surprisingly diverse.
So it goes.
Well, if you are speaking about an equilibrium mixture of two pure substances, then you would not have an emulsion, since an emulsion of two such substances would not be thermodynamically stable. Formation of an emulsion requires an additional energy input to form the unstable dispersion.
The stable structure (for instance a sphere or two side-by-side solutions) is a question for surface chemistry. Insolubility implies that the surface tension between the fluids is greater than zero (molecules in the pure liquids have a lower free energy than at the liquid-liquid interface). The result is that the liquids will attempt to minimize contact with each other. If the densities are identical, this will result not in an emulsion but in the liquids occupying side-by-side volumes, provided (as Ivan's answer points out) contact with the walls is more favorable than between the fluids, and the tension between each fluid and the container is identical. If the tension of one of the fluids in contact with the wall is lower than that of the other, then the second liquid will form a volume with a minimum surface enclosed by the first liquid (as Ivan points out).
Also, lava lamps are a fun illustration of the physical chemistry at work. Note that here thermal energy and mechanical mixing lead to fusion and breakage of globs of one liquid suspended in another. The surface tension of the glob-forming liquid in contact with the lamp walls is higher than that of the other liquid, which explains why it forms suspended globs. If the globs had a lower tension than the other liquid when in contact with the container walls, they would tend to partition to the walls (although shear due to fluid motion might keep them suspended). The density of the liquids is kept in flux, transitioning via heating and cooling through a density equivalence point that causes the globs to rise and fall.