The main goal in distributed symmetry-breaking is to understand the locality of problems; i.e., the radius of the neighborhood that a node needs to explore in order to arrive at its part of a global solution. In this work, we study the locality of matching problems in the family of regular graphs, which is one of the main benchmarks for establishing lower bounds on the locality of symmetry-breaking problems, as well as for obtaining classification results. For approximate matching, we develop randomized algorithms to show that $(1 + ε)$-approximate matching in regular graphs is truly local; i.e., the locality depends only on $ε$ and is independent of all other graph parameters. Furthermore, as long as the degree $Δ$ is not very small (namely, as long as $Δ\geq \text{poly}(1/ε)$), this dependence is only logarithmic in $1/ε$. This stands in sharp contrast to maximal matching in regular graphs which requires some dependence on the number of nodes $n$ or the degree $Δ$. We show matching lower bounds for both results. For maximal matching, our techniques further allow us to establish a strong separation between the node-averaged complexity and worst-case complexity of maximal matching in regular graphs, by showing that the former is only $O(1)$. Central to our main technical contribution is a novel martingale-based analysis for the $\approx 40$-year-old algorithm by Luby. In particular, our analysis shows that applying one round of Luby's algorithm on the line graph of a $Δ$-regular graph results in an almost $Δ/2$-regular graph.