What is the best way to reduce measurement biases when studying topics that can evoke insecurity? I'm interested in measurement biases relating to self reporting in surveys.

To add more context:

Under some circumstances, people have been known to add a positive spin when surveyed regarding goals which have significant social status attached to them. One example is how survey subjects over report the number of sexual partners they have or the amount of money they make. 

Correct me if I'm wrong, but I would assume that if someone is struggling to reach the social standard (ie: having less than what they think is the average number of sexual partners) it will be even harder to get them to be honest, since they are trying to conceal their insecurities to the researchers.

Under what experimental conditions are these people most likely to be honest? Are there any good studies out there that shed light on this question?
This isn't the exact answer to your question, but one way in which you can assess a person's likelihood of answering a measure honestly is the Social Desirability Scale (SDS). The SDS measures a participant's tendency to self-report responses that are viewed positively by society. Multiple versions of the SDS exist- here's an example of one.