A matching is a set of edges with no shared endpoints. We study how participants can retain useful matching structure when each knows its own relationships but those relationships are sensitive to outside observers. Our protocols publish a pure edge-differentially-private record of coordination messages. Each participant combines that record with its private neighbor list to obtain a shortlist of relationships; the two endpoints agree on every selected relationship. For a graph with $n$ vertices, the main protocol uses $O(n)$ communication rounds, and its completed public record can be stored in $O(n\log n)$ bits. Let $0<\varepsilon\le1$ be the privacy budget, $0<ρ<1/4$ the allowed failure probability, and $2+η$, with $0<η\le1$, the approximation factor. For a universal constant $c_0$, a per-vertex capacity of at least $c_0\log(n/ρ)/(η\varepsilon)$ suffices for two guarantees to hold together with probability at least $1-ρ$: the selected graph retains a $1/(2+η)$ fraction of the best edge count under the same capacity, and it contains an ordinary matching of at least that fraction of the original maximum. The key property is that every omitted edge touches a vertex whose shortlist is nearly full. We also prove lower bounds explaining why local decoding and a probabilistic capacity guarantee are needed, characterize a class of consistent local decoders, and give parallel and insertion-only streaming extensions. The released public messages satisfy differential privacy; the selected relationships remain a secret at their endpoints.

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