We consider word complexity and topological entropy for random substitution subshifts. In contrast to previous work, we do not assume that the underlying random substitution is compatible. We show that the subshift of a primitive random substitution has zero topological entropy if and only if it can be obtained as the subshift of a deterministic substitution, answering in the affirmative an open question of Rust and Spindeler [Indag. Math. (N.S.) 29 (2018), pp. 1131–1155]. For constant length primitive random substitutions, we develop a systematic approach to calculating the topological entropy of the associated subshift. Further, we prove lower and upper bounds that hold even without primitivity. For subshifts of non-primitive random substitutions, we show that the complexity function can exhibit features not possible in the deterministic or primitive random setting, such as intermediate growth, and provide a partial classification of the permissible complexity functions for subshifts of constant length random substitutions.