Things can differ as much as you like even if all maps are idempotents. Let F be any field. Let M be the monoid of all constant maps on {1,...n} (acting on the left) together with the identity. In otherwords, M has n left zeroes and an identity. Let V be the regular left FM-module. It is a cyclic module generated by the identity of M.
Consider the dual module V*. It can be identified with the mappings $M\to F$ with right FM-module structure given by (fm)(x)=f(mx). In particular, if m is not the identity map, the then fm is a constant map. Thus V*/constants is an n-dimensional module annihilated by all non-zero elements of M and hence cannot be generated by fewer than n elements (that is, a basis). Thus V* cannot be generated by fewer than n elements.
