Somebody leads. Different people can lead different songs if you like, but figure it out before the gig. There's nothing worse than a bunch of musicians looking at each other like deer in the headlights, waiting for somebody, anybody to count them in. Do not ask me how I know this! I got so fed up with one group that I made buttons saying:
I'M THE ONE WHO CAN COUNT TO 4
and handed one out before every gig. The leader must be fearless, ready to jump in and count the beats and name the chords like Dusty says. "Back to the A part, three, four, here we go..." Leading a group is a skill that takes practice, like any other skill. Following a leader is also a skill to practice. This could be a goal for club meetings.
One option is to get a bass player... or a drummer (one drummer)... to keep the tempo. You need an instrument with a sound that stands out from the rest of the group so everyone can hear it. A banjo (or a dulci-banjo) might do the trick. If all else fails, sit someone down in the middle of the group with a 5-gallon plastic bucket and pound the beat. Then the problem is to find the right bass/drum/banjo player. I've been stuck with bassists who can't keep a steady rhythm and bassists who play the wrong rhythm (this is a waltz, you idiot). If the bass has the wrong tempo there's nothing anyone else can do to save the tune because in a group with 3 dulcimers and one bass, the bass wins.
Once you get more than 6 MD players, as Bill describes, I think you've reached the point where somebody has to conduct. If one side is getting ahead of the other, that means they can't all hear each other. Which means they need a visual indication -- tapping foot, waving hand. I think pulling out a conductor's baton would be a funny bit of shtick for the audience.
I have played gigs where we had to watch each others' strumming hands because nobody could hear the beat. That doesn't work so well for beginners who still need to look at their fretboards, and it's hard to manage if people are using tab. At least be sure you're sitting close together and in a semi-circle so you can all see each other.