The keywords here are "randomized response". There are some interesting variations (from the Wikipedia page):

The sensitive question is worded in two dichotomous alternatives, and chance decides, unknown to the interviewer, which one is to be answered honestly.

Alternative 1: "I have consumed marijuana." Alternative 2: "I have never consumed marijuana." The interviewed are asked to secretly throw a die and answer the first question only if they throw a 6, otherwise the second question.

Also, it occurs to me that this is essentially an application of Bayes' Theorem. In an ordinary survey, the posterior probability (killed leopard|says yes) is 1, which is bad for the farmers, so they lie and therefore decrease the conditional probability (says yes|killed leopard), which is bad for the surveyors. Adding the die roll increases the unconditional probability of saying yes, so that the posterior probability no longer equals the conditional, and they can both get what they want.

I reinvented this method, except using a coin flip, during an urban economics class. We were going to actually conduct surveys in the Detroit area and thus had to learn about the problems with surveys. However my professor didn't seem very excited and pointed out some incentive shortcomings (I was too disappointed to listen too closely), and I got the impression that this is a known method with known limitations.

Suddenly, the number of "yes" responses to the leopard question started coming up by more than just one-sixth.

I fear any actual "leopard question" will not be as good as the ones that I imagined on reading that sentence.

Especially as the survey was about poaching in African countries :(