We study correlation functions of the probabilistic Schwarzian Field Theory. We compute cross-ratio correlation functions exactly in the case when the corresponding Wilson lines do not intersect, confirming predictions made in the physics literature via limit of the conformal bootstrap and the DOZZ formula. Moreover, we prove that these correlation functions characterise the measure uniquely. We use them to define and compute the stress-energy tensor correlation functions, and demonstrate, in particular, that these agree with the results obtained earlier by formal differentiation of the partition function.