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In that case though, you would know that the "No" friends are definitely not dogs, and the "Yes" friends are possibly a dog, so it seems like the dogs would still not be completely anonymous. Wouldn't the dogs be better off not partaking in the survey and being narrowed down into a group of possible dogs?

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The implementation of this should give a random answer when not being truthful.

If the coin comes you heads you answer truthfully. If it comes up tails, you flip the coin again and answer if the yes if the coin is heads and no if the coin is tails. You can then no longer know if anybody is (or is not) a dog.

The probabilities can be adjusted to provide more or less privacy (while making the data less or more useful). For example, if you only answer truthfully 0.1% of the time it would be hard to know anything about anyone, at the cost of knowing the total number of dogs less precisely.

This often helps tracking people's opinions indeed. "Here's a neat trick: If you want to work out whether your favorite celebrity is a republican, just Google their name and see if they talk about politics. If the answer is no, then they're a Republican."

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