In many fields, and especially in the medical and social sciences and in recommender systems, data are gathered through clinical studies or targeted surveys. Participants are generally reluctant to respond to all questions in a survey or they may lack information to respond adequately to some questions. The data collected from these studies tend to lead to linear regression models where the regression vectors are only known partially: some of their entries are either missing completely or replaced randomly by noisy values. In this work, assuming missing positions are replaced by noisy values, we examine how a connected network of agents, with each one of them subjected to a stream of data with incomplete regression information, can cooperate with each other through local interactions to estimate the underlying model parameters in the presence of missing data. We explain how to adjust the distributed diffusion through (de)regularization in order to eliminate the bias introduced by the incomplete model. We also propose a technique to recursively estimate the (de)regularization parameter and examine the performance of the resulting strategy. We illustrate the results by considering two applications: one dealing with a mental health survey and the other dealing with a household consumption survey.