We define and study coisotropic structures on morphisms of commutative dg algebras in the context of shifted Poisson geometry, i.e. Pn-algebras. Roughly speaking, a coisotropic morphism is given by a Pn+1-algebra acting on a Pn-algebra. One of our main results is an identification of the space of such coisotropic structures with the space of Maurer--Cartan elements in a certain dg Lie algebra of relative polyvector fields. To achieve this goal, we construct a cofibrant replacement of the operad controlling coisotropic morphisms by analogy with the Swiss-cheese operad which can be of independent interest. Finally, we show that morphisms of shifted Poisson algebras are identified with coisotropic structures on their graph.