The aim of this note is to investigate the agreement between state and parameter estimates for a class of infinite dimensional systems that utilize distributed adaptive filters. It is assumed that the unknown parameters take the form of a structured perturbation with an unknown constant parameter, and the nominal system satisfies an infinite dimensional analog of positive real lemma. The proposed adaptive observers generated by agents follow the structure of adaptive identifiers for infinite dimensional systems with the added feature of a penalty term in both the state and parameter estimates. Unlike earlier efforts, the proposed adaptive laws include a penalty term of the mismatch between the parameter estimates generated by the other agents. Conditions that provide uniform boundedness of the estimator dynamics and the parameter estimates are examined. Additionally, the effects of these modifications on the agreement amongst the state and parameter estimates are studied. The convergence of the deviation from the mean estimate is established as a metric for the performance of the consensus filters for both state and parameter estimates. The effects of the proposed consensus enforcement are illustrated with a diffusion partial differential equation example, which yield convincing numerical findings.