A valuable metric in understanding infectious disease dynamics is the local time-varying reproduction number, i.e. the expected number of secondary local cases caused by an infected individual. Accurate estimation of this quantity requires distinguishing cases arising due to local transmission from those imported from elsewhere. Methods exist to estimate the instantaneous reproductive number when cases are accurately denoted as imported or locally acquired. In practice, however, identification of cases as local or imported depends on imperfect data, such as contact tracing or genetic sequencing, leading to potential misclassification of local cases. We study the propagation of such errors on estimates of the local time-varying reproduction number. In addition, we propose a Bayesian framework for estimation of the true local time-varying reproduction number when identification errors exist. We illustrate the practical performance of our estimator through simulation studies and with outbreaks of COVID-19 in Hong Kong and Victoria, Australia.