Let's look at the Greek text (TR 1894):
εκαστος γας το ιδιον δειπνον προλαμβανει εν τω φαγειν και ος μεν πεινα ος δε μεθυει
The two words I've highlighted are men and de. When appearing close together in a statement, they constitute a contrast which means "on the one hand" and "on the other hand." (1)
In the KJV this is translated: "For in eating every one taketh before other his own supper: and one is hungry, and another is drunken."
There is only one group, the Corinthian congregation which comes together for a communal supper. Apparently, each individual (or family) brought its own food and did not share. Consequently, one the one hand, some went hungry. On the other hand, some got drunk.
Now this meal, improperly observed as it was, was in some way related to the Lord's Supper, for Paul goes on in the following verses (1 Cor. 11:23-31) to describe the proper way of observing the Lord's Supper.
Because Jesus instituted the Communion service within the context of an actual supper (see Matt. 26:28 - whether is was the Passover supper or not is disputed) the early Church combined the Communion with a meal, and this meal came to be known as the Agape. The Eerdmans Bible Dictionary (2) states the following:
The term is used in a technical sense in the New Testament to indicate the love feasts (cf. Jude 12) of the early Christians, communal meals which provided religious fellowship and were a means of charity to the poor, widowed, and orphaned of the community. . . .
Apparently the love feast was originally associated with the Eucharist or Lord's Supper (Acts 2:42, 46; 20:11; cf. 1 Cor. 11:20-21), but by the second century the two had become distinct observances.
It may have been Paul's very instructions to the Corinthians which resulted in a separation between the Agape and the Lord's Supper, due to the abuse of the former. Paul concluded his instructions: "Wherefore, my brethren, when ye come together to eat, tarry one for another. And if any man hunger, let him eat at home; that ye come not together unto condemnation. And the rest will I set in order when I come." (1 Cor. 11:33-34, KJV)
So to answer your question: There was one group, but individuals within that group behaved very differently.
(1) See Machen, J. Gresham. New Testament Greek for Beginners. The MacMillan Company. Toronto, Canada. 1923. Page 263.
(2) Revised edition by Allen C. Myers. Wm. B. Eerdmans Publishing Co. Grand Rapids, MI. 1987. Page 26.