In this paper, we study Boolean BI Logic (BBI) from a semantic perspective. This logic arises as a logical basis of some recent separation logics used for reasoning about mutable data structures and we aim at proposing new results from alternative semantic foundations for BBI that seem to be necessary in the context of modeling and proving program properties. Starting from a Kripke relational semantics for BBI which can also be viewed as a non-deterministic monoidal semantics, we first show that BBI includes some S4-like modalities and deduce new results: faithful embeddings of S4 modal logic, and then of intuitionistic logic (IL) into BBI, despite of the classical nature of its additive connectives. Moreover, we provide a logical characterization of the observational power of BBI through an adequate definition of bisimulation