This paper extends previous work on the identification of search models in which observed worker productivity is imperfectly observed. In particular, it establishes that these models remain identified even when employment histories are left-censored (i.e. we do not get to follow workers from their initial job out of unemployment), as well as when workers set different reservation wages from one another. We further show that allowing for heterogeneity in reservation can affect the empirical estimates we obtain, specifically estimates of the rate at which workers receive job offers.