You're probably thinking of a condemned criminal being sentenced to gladiatorial combat which never really happened outside of Damnatus ad ludum.

Damnatus ad ludum is when a criminal could be sentenced to gladiatorial combat. But this wasn't considered a death sentence, per se. It was basically sentencing them to slavery. They would have to serve a period of time (typically 5 years) as a gladiator. But they received training and were treated pretty much the same as any gladiator. If they survived the 5 years they were free, although it may have been possible for them to purchase their freedom earlier than that.

A condemned criminal sentenced to the arena would normally be unarmed, tied up, and facing not gladiators, but animals (Damnatio ad bestias). There would be no survivors. Here's a mosaic from 200 AD, and here's another. As you can see, the criminal is tied to a stake, held in place by at least one person and the beasts (generally large cats such as leopards and lions) may have been prodded by spears until they attacked, although most accounts just say that the animals were starved before being released.

There is a story one one criminal who was granted his freedom. Supposedly Androclus, a runaway slave, took shelter in a cave, in which a wounded lion was also sheltering. Androclus removed a thorn from the lion's paw, after which the lion became tame. They lived together for 3 years, sharing food with each other. After 3 years Androclus returned to civilization, but was recognized as a runaway slave, captured and sentenced to Damnatio ad bestias. In the arena he was set to face a newly captured lion, which just happened to be the same one he had been living with for 3 years. Instead of devouring him, the lion greets him as a friend and displays affection. Emperor Caligula, after questioning Androclus, pardons him, sets him free, and gives him the lion to keep as well. This was supposedly witnessed by a writer named Apion, who set the deed to paper. Apion's writings no longer exist, but the story is quoted in several other works which did survive, notably by Aulus Gellius in his Noctes Atticae. The story ends with:

"Afterwards we used to see Androclus with the lion attached to a slender leash, making the rounds of the tabernae throughout the city; Androclus was given money, the lion was sprinkled with flowers, and everyone who met them anywhere exclaimed, "This is the lion, a man's friend; this is the man, a lion's doctor"

How much of this story is truth and how much is fiction I'll leave for you to decide.

There are some exceptions to Damnatio ad bestias, especially for naval battle recreations (naumachia). These required thousands of people on each side. Typically the participants would be condemned criminals and captured war combatants. In these cases the survivors may have been granted freedom, but it wasn't necessarily so. Actually, there are very few notes on what became of the survivors. Presumably it was up to the Emperor whether to set them free or condemn them to death or slavery.

Cassius Dio, Epitome of Roman History:

In the Circus [in 89 CE], for example, he [Domitian] exhibited battles of infantry against infantry and again battles between cavalry, and in a new place he produced a naval battle. At this last event practically all the combatants and many of the spectators as well perished.

So other than the line "practically all the combatants and many of the spectators as well perished" there's no indication of what happened to the survivors.

Cassius Dio again, this time talking about Claudius in 52 AD. This combat reportedly involved 19,000 men:

First they assembled in a single body and all together addressed Claudius in this fashion: “Hail, Emperor! We who are about to die salute you.” And when this in no way helped save them and they were ordered to fight just the same, they simply sailed through their opponents’ lines, injuring each other as little as possible. This continued until they were forced to destroy one another.

Again it is not mentioned what became of the survivors, but given their poor performance, I'm sure that Claudius wasn't in any mood to be lenient.

Damnatus ad gladium means sentenced to death by the sword. But these were simple executions. The criminal was not given a sword to defend themselves. It was a beheading.