The randomized response technique (RRT), including the related-question RRT of Warner (1965) and the unrelated-question RRT of Greenberg et al. (1969), has been utilized to reduce underreporting of sensitive characteristics in survey studies by enhancing privacy protection. Currently, the RRT is mainly applied for prevalence estimation of some sensitive event. This work extends the application of the RRT to the analysis of time-to-event outcome. Since event time data collected from surveys are usually subject to case-I interval censoring so that only "current status" data regarding the occurrence of the event by the examination time are available, we focus on the current status (case-I interval censored) event time data collected by the RRT. Based on such data, we propose a semiparametric maximum likelihood estimation procedure for the event time distribution given the covariates, which is assumed to follow a general class of semiparametric transformation models characterized by a parametric function for the relationship between the event time and the covariates, as well as an unspecified baseline function. The asymptotic theory, including the consistency and asymptotic normality, for the proposed estimation is developed, and its finite sample properties are examined by simulation studies. We apply the proposed method to a set of current status data surveyed by the RRT to make statistical inferences on the time to the incidence of extramarital relations since marriage.