We present coarse-grained molecular dynamics simulations of polymer-mediated adhesion between chemically heterogeneous surfaces. Our surface models exhibit weakly and strongly absorbing sites in 1:1 proportion but are characterized by different degrees of segregation among these sites. When the surfaces are pulled apart, we observe systematic variations in the stress-strain curves, indicating a significant weakening of the adhesive layer on moving from finely interdispersed to more segregated morphologies. In our model systems, the macroscopic failure of the sandwiched polymer films always appears to be cohesive but, at the nanoscale, there is, in fact, a gradual transition from a cohesive to a mixed cohesive-adhesive mechanism.