All models for incomplete data either explicitly make assumptions about aspects of the distribution of the unobserved outcomes, given the observed ones, or at least implicitly imply such. One consequence is that there routinely exist a whole class of models, coinciding in their description of the observed portion of the data but differing with respect to their "predictions" of what is unobserved. Within such a class, there always is a single model corresponding to so-called random missingness, in the sense that the mechanism governing missingness depends on covariates and observed outcomes, but given these not further on unobserved outcomes. We employ these results in the context of so-called shared-parameter models where outcome and missingness models are connected by means of common latent variables or random effects, to devise a sensitivity analysis framework. Precisely, the impact of varying unverifiable assumptions about unobserved measurements on parameters of interest is studied. Apart from analytic considerations, the proposed methodology is applied to assess treatment effect in data from a clinical trial in toenail dermatophyte onychomycosis. While our focus is on longitudinal outcomes with incomplete outcome data, the ideas developed in this paper are of use whenever a shared-parameter model could be considered.