The Lefschetz properties are desirable algebraic properties of graded artinian algebras inspired by the Hard Lefschetz Theorem for cohomology rings of complex projective varieties. A standard way to create new varieties from old is by forming connected sums. This corresponds at the level of their cohomology rings to an algebraic operation also termed a connected sum, which has recently started to be investigated in commutative algebra by Ananthnarayan-Avramov-Moore. It is natural to ask whether abstract algebraic connected sums of graded Gorenstein artinian algebras enjoy the Lefschetz properties in the absence of any underlying topological information. We investigate this question as well as the analogous question concerning a closely related construction, the fibered product.