We present a number of algebraic results that facilitate the study of the deductive interpolation property and closely-related metalogical properties for algebraizable deductive systems. We prove that, when V is a congruence-distributive variety, V has the congruence extension property if and only if the class of finitely subdirectly irreducible members of V has the congruence extension property. When a deductive system is algebraized by V, this provides a description of local deduction theorems in terms of algebraic models. Further, we prove that for a variety V with the congruence extension property such that the class of finitely subdirectly irreducibles is closed under subalgebras, V has a one-sided amalgamation property (equivalently, since V is a variety, the amalgamation property) if and only if the class of finitely subdirectly irreducibles has this property. When V is the algebraic counterpart of a deductive system, this yields a characterization of the deductive interpolation property in terms of algebraic semantics. We announce a similar result for the transferable injections property, and prove that possession of all these properties is decidable for finitely generated varieties satisfying certain conditions. Finally, as a case study, we describe the subvarieties of a notable variety of BL-algebras that have the amalgamation property.