Probably the most enjoyable and most worthwhile regular group of which I have been a member was the Rambler Club in the School of English at Durham University, late 1980s.
The name came from the fact that the group was focused on reading one essay per week from the 208 essays by Samuel Johnson published as a periodical called The Rambler from 1750-52. I was lucky enough to be asked to join for the last 20 or so weeks, completing the sequence - I infer the group had previously been meeting for at least seven years (since the group met only during term time, which lasted 27 weeks but some weeks would be missed to due exams). After finishing Johnson's Rambler, I believe they moved on to the Adventurer and Idler essay sequences.
There were four 'core' members (which seems common for many lasting groups) of common interest. The 'Chair' was Derek T, whose characteristics were a nimbleness of wit, fertility of ideas, and a crisp phraseology - he tended to talk the most and shape the debate. The second most frequent speaker was David C - who was the most open-hearted and emotional speaker - he would also tend to give the conversation a dark and pessimistic turn. David F spoke only when he had something considered to say - with some diffidence, but always respectfully listened-to because his statements had a background of deep thought. Tom C was the oldest and most distinguished member, he beamed upon proceedings with a benign air - and he was turned-to to settle disagreements of fact, or he would chip-in with a 'crowning' verbatim quotation from a memory exceptionally well-stocked with the classics; especially Shakespeare (upon whom he was a great authority).
The structure of each meeting was quite simple. The timing was about an hour, people arrived carrying a packed-lunch, either having read the essay already - or, if not, then being given a photocopy to read while others arrived. Then the conversation would be kicked-off by Derek, who would usually take charge of moving it on or redirecting it as necessary.
The Rambler essays were essentially a stimulus to conversation, and the conversation was 'moral' in theme - typically beginning with whatever moral point the essay had emphasised, but evolving unpredictably according to the mind of the group and their interactions.
And the conversations were superb - due to the quality of the participants - especially the informal chairmanship of Derek T, and the necessary degree of structure. Behind the formal structure - and, I think vital to the success for the group - was a common purpose or philosophy; which was 'anti-critical'. These were hard-working and experienced teachers of English in one of England's premier universities who yet were very sceptical of the validity and value of mainstream 'literary criticism'; and were seeking a more personal, heart-felt and spontaneous way of discussing literature. By my estimation, they achieved it.
I have attended and tried to form many small conversation groups, sporadically over the years - and they are in my experience, seldom at a high level and always very difficult to sustain - so I feel privileged to have participated in one of the shining exceptions; albeit briefly.