Abstract: Given a data set of size $n$ in $d'$-dimensional Euclidean space, the $k$-means problem asks for a set of $k$ points (called centers) such that the sum of the $\ell_2^2$-distances between the data points and the set of centers is minimized. Previous work on this problem in the local differential privacy setting shows how to achieve multiplicative approximation factors arbitrarily close to optimal, but suffers high additive error. The additive error has also been seen to be an issue in implementations of differentially private $k$-means clustering algorithms in both the central and local setting (Balcan et. al. (2017), Chaturvedi et. al. (2021), Chang et. al. (2021)). In this work we introduce a new locally private $k$-means clustering algorithm that achieves near-optimal additive error whilst retaining constant multiplicative approximation factors and round complexity. Concretely, given any $c>\sqrt{2}$, our algorithm achieves $O(k^{1 + \tilde{O}(1/(2c^2-1))} \sqrt{d' n} \mbox{poly} \log n)$ additive error with an $O(c^2)$ multiplicative approximation factor.