Proper class sized monster models are typically formulated in a class theory like $NBG$ and they can reasonably be formalized in $ZFC$ with some kind of global choice, but for some theories you don't need either classes as a first-order object or global choice to exhibit proper class sized monster models.

For a complete first-order theory $T$, a class model, $\mathfrak{C}$, is a class $C$ (as in the class of sets satisfying some formula maybe with parameters) with relations and functions also given by formulas maybe with parameters, such that $\mathfrak{C}$ models $T$ in the obvious way (although there is a subtlety here with $ZFC$ proving that $\mathfrak{C}$ models $T$ uniformly vs. individually proving that it models each finite set of sentences in $T$). A proper class monster model is a class model $\mathfrak{C}$ such that $C$ is a proper class and which is 'proper class saturated,' i.e. for every sub*set* $A\subset C$, $\mathfrak{C}$ realizes every type in $S_1(A)$ (I'm not going to worry about homogeneity right now).

For uncountably categorical theories and other extremely nice theories there clearly are proper class monster models. For an uncountably categorical theory we can take an Ehrenfeucht-Mostowski model with $Ord$ for its spine (which, it should be noted, at least gives us the existence of a proper class model for any theory, which isn't a priori obvious). You should be able to do something similar with $\omega$-stable $\omega$-categorical theories in light of the coordinatization theorem (there's a finite collection of totally categorical strongly minimal sets that models of this theory are prime over). I'm pretty sure that if you take a class sized model of a unidimensional theory and take an appropriate ultrapower of it (which is well-defined for set sized index sets, using the index set and ultrafilter as parameters) that the ultrapower will be proper class saturated.

It is plausible that the same can be done for stable theories in general (or at least nice enough stable theories), although I haven't tried to think about the details. On the other hand there are a couple of amusing examples of unstable theories with simple to describe proper class monster models. The surreal numbers are a proper class monster model of $RCF$ (and therefore $DLO$ in the reduct) and if we define an edge relation on the class of all sets by $xEy$ iff $x\in y$ or $y\in x$, then we get a proper class monster model of the theory of the random graph.

Is there a general method for exhibiting proper class monster models in $ZFC$ without global choice? If not are there nice families of theories for which we can exhibit proper class monster models? Is there a model $V$ of $ZFC$ and a theory $T$ that does not have a proper class monster model in $V$?