The Mallows measure is measure on permutations which was introduced by Mallows in connection with ranking problems in statistics. Under this measure, the probability of a permutation $π$ is proportional to $q^{Inv(π)}$ where $q$ is a positive parameter and $Inv(π)$ is the number of inversions in $π$. We consider the length of the longest common subsequence (LCS) of two independently permutations drawn according to $μ_{n,q}$ and $μ_{n,q'}$ for some $q,q' >0$. We show that when $0<q,q'<1$, the limiting law of the LCS is Gaussian. In the regime that $n(1-q) \to \infty$ and $n(1-q') \to \infty$ we show a weak law of large numbers for the LCS. These results extend the results of \cite{Basu} and \cite{Naya} showing weak laws and a limiting law for the distribution of the longest increasing subsequence to showing corresponding results for the longest common subsequence.