We study (coalitional) exchange stability, which Alcalde [Economic Design, 1995] introduced as an alternative solution concept for matching markets involving property rights, such as assigning persons to two-bed rooms. Here, a matching of a given Stable Marriage or Stable Roommates instance is called coalitional exchange-stable if it does not admit any exchange-blocking coalition, that is, a subset S of agents in which everyone prefers the partner of some other agent in S. The matching is exchange-stable if it does not admit any exchange-blocking pair, that is, an exchange-blocking coalition of size two. We investigate the computational and parameterized complexity of the Coalitional Exchange-Stable Marriage (resp. Coalitional Exchange Roommates) problem, which is to decide whether a Stable Marriage (resp. Stable Roommates) instance admits a coalitional exchange-stable matching. Our findings resolve an open question and confirm the conjecture of Cechlárová and Manlove [Discrete Applied Mathematics, 2005] that Coalitional Exchange-Stable Marriage is NP-hard even for complete preferences without ties. We also study bounded-length preference lists and a local-search variant of deciding whether a given matching can reach an exchange-stable one after at most k swaps, where a swap is defined as exchanging the partners of the two agents in an exchange-blocking pair.