We discuss the analytic structure of off-shell correlation functions in Little String Theories (LSTs) using their description as asymptotically linear dilaton backgrounds of string theory. We focus on specific points in the LST moduli space where this description involves the spacetime Rd-1,1×SL(2)/U(1) times a compact CFT, though we expect our qualitative results to be much more general. We show that n-point functions of vertex operators O(p μ) have single poles as a function of the d-dimensional momentum pμ, which correspond to normalizable states localized near the tip of the SL(2)/ U(1) cigar. Additional poles arise due to the non-trivial dynamics in the bulk of the cigar, and these can lead to a type of UV/IR mixing. Our results explain some previously puzzling features of the low energy behavior of the Green functions. As another application, we compute the precise combinations of single-trace and multi-trace operators in the low-energy gauge theory which map to single string vertex operators in the N=(1,1) supersymmetric d=6 LST. We also discuss the implications of our results for two-dimensional string theories and for the (non-existence of a) Hagedorn phase transition in LSTs.