We derive equivalent conditions for the (local) absolute continuity of two laws of semimartingales on random sets. Our result generalizes previous results for classical semimartingales by replacing a strong uniqueness assumption by a weaker uniqueness assumption. The main tool is a generalized Girsanov's theorem, which relates laws of two possibly explosive semimartingales to a candidate density process. Its proof is based on an extension theorem for consistent families of probability measures. Moreover, we show that in a one-dimensional It��-diffusion setting our result reproduces the known deterministic characterizations for (local) absolute continuity. Finally, we give a Khasminskii-type test for the absolute continuity of multi-dimensional It��-diffusions and derive linear growth conditions for the martingale property of stochastic exponentials.