The trouble is that we have no idea how Greek comedy was received by its immediate audience, except that we are sometimes told which one won the prize for that year. We simply do not know whether the audience "got" the jokes - if we even know what the jokes were. The text of eleven Old Comedy plays survives; audience reactions to them do not.

In theory there were no restrictions on who could see a play. In practice it depended on who had the leisure time to travel to the theatre/festival where they were performed. This may have meant that the audience predominantly consisted of wealthier folk, but it is very difficult to distinguish between social classes in Classical Greece, and there was nothing like what you call a "bourgeoisie sphere". Even the views of the author himself are open to interpretation, because we cannot tell whether he backed the views his comedies presented, or whether he merely played them for laughs.

In terms of differences, actually Old Comedy looks fairly familiar to anyone who knows his modern slapstick and political satire. There's fart jokes, rude words, caricatures, cross-dressing and tons of innuendo. Prominent figures from politics and society are ruthlessly mocked and ridiculed. It's all the same stuff - we just hope we're getting it right when we make claims about what's being made fun of, why, and how.

If you're speaking of classical (5th century BCE) Greece, then the answer is something people are arguing over still. For example, many historians used to think that about 20,000 people attended the main Dionysia festival in Athens, which would mean that it was VERY open. Then that figure got revised to closer to 15,000, and there is a growing number of historians and classicists who are thinking it's more like 5,000, which would imply it's a bit less open. Still, 5,000 is a large number in an ancient city!

If we're still talking about the City Dionysia (a major festival mostly consisting of dancing, tragedy, and comedy), then no one can agree on the composition of the audience, in particular if women were able to attend. However, theoretically, if you were a free male, you could attend, and Pericles instituted a fund from public money to buy tickets for people who could not afford to go. There is vague evidence to suggest that people from many occupations and social levels attended the plays.

This gets more complicated because there was also a comic festival in Athens every year and festivals in the small communities/suburbs around Athens that we know very little about.

If you're asking specifically about satire, you're asking about Satyr Plays, which were often done after a trilogy of tragedies during the Dionysia. These were often funny little plays that retold a classical myth but with a chorus of alcoholic, horny goat people running around causing a mess. The main differences would have been that everyone wore masks, there was a chorus, and lots of singing. Still, as Iphikrates, fart jokes are universal. The plots tend to be very different, but they rely on a lot of comic devices that we still look to today.

For a source on the audiences at the City Dionysia, the CAMBRIDGE COMPANION TO GREEK TRAGEDY has a good piece by Simon Goldhill called "The Audience of Athenian Tragedy" and Edith Hall "The Sociology of Athenian Tragedy."
(https://openlibrary.org/works/OL16952548W/The_Cambridge_companion_to_Greek_tragedy)

For more recent debates in the field, see Melinda Powers's ATHENIAN TRAGEDY IN PERFORMANCE