Recently, a series of papers proposed deep learning-based approaches to sample from unnormalized target densities using controlled diffusion processes. In this work, we identify these approaches as special cases of the Schrödinger bridge problem, seeking the most likely stochastic evolution between a given prior distribution and the specified target, and propose the perspective from measures on path space as a unifying framework. The optimal controls of such entropy-constrained optimal transport problems can then be described by systems of partial differential equations and corresponding backward stochastic differential equations. Building on these optimality conditions and exploiting the path measure perspective, we obtain variational formulations of the respective approaches and recover the objectives which can be approached via gradient descent. Our formulations allow us to introduce losses different from the typically employed reverse Kullback-Leibler divergence that is known to suffer from mode collapse. In particular, we propose the so-called log-variance loss, which exhibits favorable numerical properties and leads to significantly improved performance across all considered approaches.