Answering this for the Roman part only, I havn't paid much attention to the Greek religion practices.

So, first of all, when talking about the average citizens and what they did or didn't do it's important to know that it's extremely difficult to tell anything for certain. There wasn't anything like Facebook or Twitter back then that would de facto prove that people were there. However, due to the organizations we can say that they most likely participated, because they had the option to do so in some cases and because the meat was shared.

I wrote a short post about the organization of the Imperial Cult a few months ago, which generelly describes the way all organized religion around gods would have worked. Étienne writes about these organizations in his book from 1958 (there is probably a better source for these priest organizations but this is where I found out about them).

A standard sacrifice was the bull and we know from imagery and text sources that it was used in sacrifices to Jupiter, Mithras and the Emperor's genius, as well as other gods (see Gradel, 2002). How often these sacrifices took place is a bit difficult to say. In the case of the Emperor's genius it would have been done on his birthday, but perhaps also on the annual day of his accession and other occassions.

What's interesting here though is that we know what happened to the meat after these sacrifices. Instead of just throwing it away it was given away to the attending people (who wouldn't normally have meat in their diet). The best parts were given to the god, but the rest of the bull was cooked and eaten. To clarify, this would be the practice in the case of a sacrifice done by the state or regional priests. It's unsure whether municipal priests would have performed these kinds of sacrifices, because the imagery we have only show imperial statues or temples (see Price 1980 for this section).

This is a quote describing a sacrifice by the Arval Brethren (which would not have been average citizens) that both gives an idea of what kind of occassion could prompt an animal sacrifice and also shows the variety of animals sacrificed:

Seven days before the Ides of November [7 November], the Arval Brothers assembled in the grove of Dea Dia [---] And there they made sacrifice because in a violent storm some trees in the sacred grove of Dea Dia were struck by lightning and burnt; [---] a purificatory sacrifice was carried out with the offering of a full-grown pig, ram and bull [suovetaurilia]. Then in front of the temple cows, their horns bound with gold, were sacrificed to Dea Dia — total 2; then at the altars built for the occasion sacrifices were made to the gods as listed below: to Janus Pater, rams — 2; to Jupiter, wethers — 2; to Mars Pater Ultor, rams — total 2; to deity, male or female, wethers — 2; to the spirit of Dea Dia, sheep — total 2; to the virgin deities, sheep — total 2; to the attendant deities, wethers — total 2; to Fons [the god of springs], wethers — total 2; to Flora, sheep — total 2; to Summanus Pater, black wethers — 2; to Vesta Mater, sheep — 2; to vesta of the gods and goddesses, sheep — 2; likewise to Adolenda and Coiquenda, sheep — 2; and before the shrine of the Caesars, to the spirit of our lord, the emperor Severus Alexander, a bull with gilded horns; likewise to the divi, totalling 20, wethers — 20 (Beard 1998, CIL VI 2107).

Sources:

Price, S.R.F. (1980), "Between Man and God: Sacrifice in the Roman Imperial Cult" in Journal of Roman Studies, vol. 70, pp. 28-43.

Gradel, I. (2002), Emperor Worship and Roman Religion.

Ètienne, R. (1958), Le Culte Impérial dans la Péninsule Ibérique d'Auguste à Dioclétien.

Beard, M. et al. (1998), Religions of Rome, vol. 2, A Sourcebook