Both, if I heard correctly from a prof.

Since many events were put on by richer citizens, attendance would be free. At least that was the case in Rome. I have little doubt that there were also performances that had a small charge to get in. Considering that theatre and sport were main attractions and entertainment, much of it could be accessed for free. Indeed at some events, there would be tesserae used as tokens for admittance, or dispensed to the crowd to redeem for everything from bread to prostitutes. These were the kinds of events that lead to Juvenal's "panem et circenses" (bread and circuses) observation. Political leaders and rich Romans would use these events to occupy the people and win over crowds.

For the Greek end, I know less.