Smartphones present many interesting opportunities for survey research, particularly through the use of mobile data collection applications (apps). There is still much to learn, however, about how to integrate apps in general population surveys. Recent studies investigating hypothetical willingness to complete mobile data collection tasks via an app suggest there may be substantial resistance, in particular, due to concerns around data privacy. There is not much evidence about how privacy concerns influence actual decisions to participate in app-based surveys. Theoretical approaches to understanding privacy concerns and survey participation decisions would suggest that the influence of the former over the latter is likely to vary situationally. In this paper, we present results from a methodological experiment conducted in the context of a three-wave probability-based online panel survey of the general population as part of the 2019 Swiss Election Study ("Selects") testing different ways of recruiting participants to an app. Questions included at wave 1 about online data privacy concerns and comfort sharing different types of data with academic researchers allow us to assess their impact on both hypothetical willingness to download a survey app for completing questionnaires, to take and share photos, and to share the smartphone's GPS location and actual completion of these tasks. Our findings confirm that general concerns about online data privacy do influence hypothetical willingness to complete mobile data collection tasks, but may be overridden by how comfortable people feel about sharing specific types of data with researchers. When it comes to actual compliance with task requests, however, neither privacy concerns nor comfort sharing data seem to matter. We conclude with recommendations for exploring these relationships further in future app-based studies.