This is one of two mildly related questions.
When $T$ is an incomplete theory with quantifier elimination, the completions of $T$ are determined by the 'characteristic' of the models (i.e. the theory of the substructure generated by $0$).
Is it possible to describe what determine the completions of a theory with model-completeness? (Sometimes it is the set of $\exists$-definable elements, but it may not be true in general.)
The reason your statement about theories with QE is true is that in a theory $T$ with QE, every sentence is equivalent to a quantifier-free sentence, so picking a completion of $T$ amounts to deciding the truth value of all the quantifier-free sentences. Semantically speaking, this amounts to describing the substructure generated by the empty set.
[Strictly, for the above statement to be true in a language without constant symbols, we have to admit empty structures (in order to have a substructure generated by the empty set), as well as the basic logical symbols $\top$ and $\bot$ (in order to have any quantifier-free sentences at all). In such a language, case every theory with quantifier elimination must be complete. I feel strongly that these conventions are the correct ones.]
Now if $T$ is a model complete theory, every sentence is equivalent to a universal sentence, so picking a completion $T'$ of $T$ amounts to deciding the truth value of all the universal sentences (and their negations, the existential sentences). The set $T'_\forall$ of all universal consequences of $T'$ is the theory of all substructures of models of $T'$, so this is equivalent on the semantic side to specifying the class of structures which appear as substructures of models of $T'$.
Since the class of models of a universal theory is always closed under directed colimits, and every $L$-structure is a directed colimit of finitely generated substructures, this is further equivalent to specifying the class of finitely generated substructures of models of $T'$. To put it another way, an existential sentence $\exists \overline{x}\, \varphi(\overline{x})$ contains information about a finite piece of the diagram of the substructure generated by the finite tuple $\overline{x}$, and a finitely generated structure is a substructure of a model of $T'$ if and only if every finite piece of its diagram is represented in $T'_\exists$ in this way.
I think this condition about specifying a class of allowable finitely generated structures is the natural generalization of the situation under QE, where you just specify a single $0$-generated structure.