Even when anonymous?
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I found this (relatively) old paper[1]. It mostly deals with reporting on government forms, but there is a little bit about sensitivity of the information reported.
Singer, Von Thurn, and Miller (1995) have shown that assurances of confidentiality
reduce nonresponse and/or increase the accuracy of responses to income questions and other
sensitive survey topics. In addition, recent technological advances raise some possibilities for
gathering income information under more anonymous conditions, without the direct involvement
of an interviewer, which may substantially reduce sensitivity pressures and thus yield more
complete reporting. Research is starting to show the benefits of anonymous reporting — for
example, via audio computer-assisted self-interviewing (A-CASI) — on the quality of sensitive
behavior reports (Turner, Lessler, and Devore 1992; Tourangeau and Smith 1996), although as
yet there has been no clear demonstration that income reports would similarly benefit.
Singer, Von Thurn, and Miller (1995)[2] concludes that assurances of anonymity affect response to questions about sensitive data, but other factors may play a role (such as trust in the interviewer, etc.), and the effect "is not large".
1 - https://www.census.gov/srd/papers/pdf/sm97-05.pdf
2 - http://apps.webofknowledge.com/full_record.do?product=WOS&se...