This is the second paper in a series of two studying monogenicity of number rings from a moduli-theoretic perspective. By the results of the first paper in this series, a choice of a generator $$\theta $$ for an A-algebra B is a point of the scheme $${\mathcal {M}}_{B/A}$$ . In this paper, we study and relate several notions of local monogenicity that emerge from this perspective. We first consider the conditions under which the extension B/A admits monogenerators locally in the Zariski and finer topologies, recovering a theorem of Pleasants as a special case. We next consider the case in which B/A is étale, where the local structure of étale maps allows us to construct a universal monogenicity space and relate it to an unordered configuration space. Finally, we consider when B/A admits local monogenerators that differ only by the action of some group (usually $${\mathbb {G}}_m$$ or $$\textrm{Aff}^1$$ ), giving rise to a notion of twisted monogenerators. In particular, we show a number ring A has class number one if and only if each twisted monogenerator is in fact a global monogenerator $$\theta $$ .