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At STOC 2006 and CRYPTO 2007, Beimel et. al. introduced a set of privacy requirements for algorithms that solve search problems. In this paper, we consider the longest common subsequence (LCS) problem as a private search problem, where the task is to find a string of (or embedding corresponding to) an LCS. We show that deterministic selection strategies do not meet the privacy guarantees considered for private search problems and, in fact, may "leak" an amount of information proportional to the entire input. We then put forth and investigate several privacy structures for the LCS problem and design new and efficient output sampling and equivalence protecting algorithms that provably meet the corresponding privacy notions. Along the way, we also provide output sampling and equivalence protecting algorithms for finite regular languages, which may be of independent interest.