We study how Hilbert bimodules correspond in the algebraic case to hermitian Morita equivalences and consequently we obtain a description of the hermitian Picard group of a commutative involutive algebraAas the semidirect product of the classical hermitian Picard group ofAand the automorphisms ofAcommuting with the involution. We also obtain similar decomposition results on hermitian Picard groups of involutive coalgebras (Cωc), which show, at least in the cocommutative case, that this hermitian Picard group differs considerably from the non hermitian one.