We study the multifractal analysis of self-similar measures arising from random homogeneous iterated function systems. Under the assumption of the uniform strong separation condition, we see that this analysis parallels that of the deterministic case. The overlapping case is more complicated; we introduce the notion of finite type for random homogeneous iterated function systems and give a formula for the local dimensions of finite type, regular, random homogeneous self-similar measures in terms of Lyapunov exponents of certain transition matrices. We show that almost all points with respect to this measure are described by a distinguished subset called the essential class, and that the dimension of the support can be computed almost surely from knowledge of this essential class. For a special subcase, that we call commuting, we prove that the set of attainable local dimensions is almost surely a closed interval. Particular examples of such random measures are analyzed in more detail.