I don't know which enzyme you mean, but the same situation is for thymidylate synthase. In fact, one monomer binds the substrate at the moment, and after the catalytic cycle, the other monomer binds also the substrate. I do not remember exactly but probably such mechanism helps to efficiently remove the products from the catalytic site. There is a binding of a substrate to the one monomer, and simultaneously there is a catalysis or even dissociation of a product on the other monomer. And both monomers work alternately. Thus, you should not see probably the negative cooperativity. In the crystal structure, you see just a time-point. In other words, you have at the moment always only one active site binding substrate and you cannot saturate the other because of subtle structural changes induced by such binding.

The alternating site catalysis mechanism described by Tomasz Fraczyk is also followed by F1-Fo ATP synthase, although with 3 active sites instead of two. A Nobel Prize was awarded to Paul Boyer for elucidating this mechanism.

You may be able to occupy the second site by adding the product instead of the substrate, although it may require quite a high concentration.

P-type ATPases like Na/K-ATPase have a similar mechanism, one protomer is in the Na-binding E1, the other in the K-binding E2-conformation. This allows coupling of the energy-generating ATP hydrolysis in E1 with the energy-requiring de-occlusion of K in the E2 protomer. Occluded Rb (as K-congener) has a half-life of 18 min, yet the turnover number of the enzyme is some 10000/min, that's how efficient the coupling is.