What you're talking about sounds like activity that is common for a contest or a QSO party. If you're not familiar with QSO parties they are organized like contests, but the organizers try to emphasize everyone having fun, and don't focus quite as much on scores. There is a saying though, that if something smells, looks, and feels like an elephant then it's probably an elephant; in other words, QSO parties are generally regarded as contests, and are included in contest calendars.
Contests and QSO parties almost always include a signal report, which makes them bona-fide contacts in the eyes of most hams. However these days the signal report is almost always "59" for phone or "599" for Morse. Were the first two digits of the six-digit number "59"? Contesters like to leave a little pause between the signal report and the "exchange", but to outsiders, "five nine, one three seven six" might sound like a six-digit number. What's an exchange, you ask? It's a piece of data that the contest rules require to be exchanged. It might be the location of the station encoded in a numeric format, or a sequential serial number, or the year the operator was first licensed, or the name of the state or province the station is in, or just about anything else the contest organizers can dream up. For some contests the exchange changes every contact, but for others it's always the same for a particular station.
As for all talking at once, contests are competitive and operators will talk over each other in an attempt to get a contact with a rare multiplier. What's a multiplier? As an example, rules for a state's QSO party might define a contester's score as the number of contacts times the number of counties contacted (usually on a per-band basis). Typically there are rare multipliers, like a sparsely-populated county in the state's QSO party, that everyone is trying to contact. It's not uncommon for several stations to try to work the same station, usually a rare multiplier, at the same time, a situation known as a pileup. A skilled operator can pick individual call signs out of the cacophony and keep calling operators happy.
Here's a typical contest exchange ("W1ABC" and "VA3XYZ" were made up for this example, but may well exist as real call signs):
VA3XYZ: victor alpha three x-ray yankee zulu QRZ
W1ABC: whiskey one alpha bravo charlie
VA3XYZ: W1ABC, five-nine, one three seven six
W1ABC: thanks, five nine, four eight seven
VA3XYZ: seventy-three QRZ
So that was a valid QSO because signal reports were exchanged, although the reports were the standard 59s, and acknowledged with "thanks" and "seventy-three". VA3XYZ repeated the calling station's call sign; the calling station W1ABC didn't repeat VA3XYZ's, but implied that he or she did, because if he or she didn't copy the call sign then he or she would have requested a repeat (a "fill" in contest jargon). They were valid QSOs for this particular imaginary contest because the exchange included serial numbers, "1376" and "487".
Does that describe what you heard? If not, the next time you hear such activity, please make a note of details such as date, time, frequency, and transcriptions of a few QSOs, and we can probably help you out. Or write down a few call signs and email the operators to ask, and then come back here and answer your own question. (Many operators list their email addresses in qrz.com.)