When separating an Oreo cookie, why does the cream stick to just one side only?

The "stuff" sticks to itself better than it sticks to the cookie. Now if you pull the cookies apart, you create a region of local stress, and one of the two interfaces will begin to unstick. At that point, you get something called "stress concentration" at the tip of the crack (red arrow) - where the tensile force concentrates:

To get the stuff to start separating at a different part of the cookie, you need to tear the stuffing (which is quite good at sticking to itself) and initiate a delamination at a new point (where there is no stress concentration).

Those two things together explain your observation.

Cookie picture credit (also explanation about manufacturing process introducing a bias)

Update

A plausible explanation was given in this article describing work by Cannarella et al:

Nabisco won’t divulge its Oreo secrets, but in 2010, Newman’s Own—which makes a very similar “Newman-O”—let the Discovery Channel into its factory to see how their version of cookies are made. The key aspect for twist-off purposes: A pump applies the cream onto one wafer, which is then sent along the line until a robotic arm places a second wafer on top of the cream shortly after. The cream always adheres better to one of these wafers—and all of the cookies in a single box end up oriented in the same direction. Which side is the stronger wafer-to-cream interface? “We think we know,” says Spechler. The key is that fluids flow better at high temperatures. So the hot cream flows easily over the first wafer, filling in the tiny cracks of the cookie and sticking to it like hot glue, whereas the cooler cream just kind of sits on the edges of those crevices.

One of the first places you might encounter tensors, at least one of the more classical applications of them, is in the mathematical theory of rheology. That is an indication that the processes of transmitting stress can't be done generally by just a vector approach. In other words, its really complicated.

The answer, as already given, is correct (as far as it goes).

If we assume there are 3 bulk objects (the two cookie wafers and the patty of filling) then the weakest with respect to tensile stress will rupture to relieve that stress. Actually, we usually consider each adhesive bond (on either side of the patty) is composed of the two bulk materials, and the two surfaces. If you were to take a high-speed video of the process, you could probably see that both of the objects deform as stress is applied. They not only deform, but react to the deformation by compression, and in the case of the patty by flow and transmission of the forces in a highly non-linear fashion.

When the adhesion between two surfaces is weaker than the bulk cohesive strength of either, we expect adhesive failure rather than cohesive (bulk) failure, as a rule of thumb. Unfortunately, that "wisdom" is only approximate. In most cases, when we actually analyze the surfaces after adhesive failure we find that there's some wafer residue on the patty and there's some patty on the wafer. Meaning it's usually (not always, but usually) in one of the interface zones (the transition sone between bulk properties and the actual atomic surface) where the failure occurred. If the adhesive force was weaker, we'd predict a "clean break" and we rarely get that (although it's often clean enough to fool our eyes).

I just looked at the picture from one of the answers, you can clearly see that wafer powder is still embedded in the white sugar paste. I'm fairly certain if you were to examine the underside of the wafer, you'd see sugar paste on it, too.

The reason adhesive failures (remember this is in contrast to cohesive failure. Gluing two balsa wood strips together and pulling them apart will probably result in cohesive failure of the wood.) The simplified reason for this is that in the transition zone, the material is neither structured like the bulk on one side, nor structured like the bulk material on the other, and this disorder means the forces in that area are even weaker than either bulk, while actual adhesive forces tend to be much greater immediately at the surfaces on an atomic scale.