In Maryland, clubs play statewide inter-club/course "A Team" matches, where courses field their 12 best players and play home-and-away fourball matches (3 teams at home, 3 teams away) in a bracket-like tournament. I think most clubs/courses participate, so you can have matches between, say, Congressional and the Falls Road muni that's just up the road. I'm not sure if these matches are sponsored by the MSGA.
My club also participates in "B Team" matches among a group of six local clubs. The matches are played round-robin style, so each club plays each other club once, and then there is a final day where all six clubs play matches at one of the clubs. It's designed to let guys who aren't good enough to play "A Team" still play inter-club matches; there are handicap minimums to be able to play. Our group is Congressional, Columbia, Chevy Chase, Lakewood, Bethesda, and Woodmont, and I've heard that a group including Four Streams, Caves Valley, and Baltimore has formed or is forming a similar group.