  We endow twisted tensor products with a natural notion of counit and
comultiplication, and we provide sufficient and necessary conditions making the
twisted tensor product a counital coassociative coalgebra. We then characterize
when the twisted tensor product of bialgebras is a bialgebra, and when the
twisted tensor product of Frobenius algebras is a Frobenius algebra. Our
methods are purely diagrammatic, so these results hold for (braided) monoidal
categories. As an application, we recover that some quantum complete
intersections are Frobenius algebras, and we construct families of
noncommutative symmetric Frobenius algebras. Along the way, we also
characterize when twisted tensor products of separable algebras are separable,
and we prove that twisted tensor products of special Frobenius algebras are
special Frobenius.
