We consider two distinctly different gauged theories of Floreanini-Jackiw type chiral boson. Using the Batalin-Fradkin-Vilkovisky (BFV) technique, the nilpotent BRST, and anti-BRST symmetry transformations for these theories have been studied. Other forms of nilpotent symmetry transformations like co-BRST, and anti-co-BRST, which leave the gauge-fixing part of the action invariant, are also explored for these two models. We show that the nilpotent charges corresponding to these symmetries for both the models satisfy the algebra of the de Rham cohomological operators in differential geometry. The Hodge decomposition theorem on the compact manifold is also studied in the context of conserved charges for these two models. It has been found that despite the differences of constraint structures and the theoretical spectra the models exhibit identical symmetry properties.