Large samples and long time series are often needed for effective broad-scale monitoring of status and trends in wild populations. Obtaining those sample sizes can be more feasible when volunteers contribute to the dataset, but volunteer-selected sites are not always representative of a population. Previous work to account for biased site selection has relied on knowledge of covariates to explain differences between site types, but such knowledge is often unavailable. For cases where relevant covariates have not been defined, we used a simulation study to identify the consequences of including non-probabilistically selected sites (NP sites) in addition to sites selected from a probability-based design (P sites), test modeling frameworks that might correct for biases, and evaluate whether those frameworks could allow NP sites to reduce the sampling requirement for P sites and potentially reduce costs of monitoring. We informed the simulation with pilot data from surveys of monarch butterflies and their obligate larval host plant, milkweed. We found strong biases in NP sites versus P sites in density and trends of monarchs and milkweed. Modeling frameworks that accounted for site type with a group effect or that strongly downweighted NP sites successfully produced unbiased estimates. However, sampling more NP sites typically did not improve accuracy or precision, and adding NP sites sometimes required also adding P sites to prevent biases. Further work on novel modeling frameworks would be useful to allow citizen-science data to contribute useful information to conservation.