Session type systems have been given logical foundations via Curry-Howard correspondences based on both intuitionistic and classical linear logic. The type systems derived from the two logics enforce communication correctness on the same class of π-calculus processes, but they are significantly different. Caires, Pfenning, and Toninho informally observed that, unlike the classical type system, the intuitionistic type system enforces locality for shared channels, i.e. received channels cannot be used for replicated input. In this paper, we revisit this observation from a formal standpoint. We develop United Linear Logic (ULL), a logic encompassing both classical and intuitionistic linear logic. Then, following the Curry-Howard correspondences for session types, we define πULL, a session type system for the π-calculus based on ULL. Using πULL we can formally assess the difference between the intuitionistic and classical type systems, and justify the role of locality and symmetry therein.