Related:What caused Gitara's death after she Foretold Rand's birth? (this also has a convenient quote you can use for this question)
We haven't witnessed many foretelling, but in at least Gitara's one about Rand, she was screaming about what she was Foretelling--Rand's birth. Does it always have to be that way? Could someone have a Foretelling without anyone else knowing if they stayed quiet?
Elaida has a Foretelling about Rand at the Palace in Caemlyn in book 1. Although her Foretelling is perhaps even more shocking and terrible than Gitara Moroso's, she not only doesn't shout but retains control of herself to the extent that she can speak quietly enough to ensure nobody but Rand hears her:
"This I Foretell," Elaida replied, "and swear under the Light that I can say no clearer. From this day Andor marches toward pain and division. The Shadow has yet to darken to its blackest, and I cannot see if the Light will come after. Where the world has wept one tear, it will weep thousands. This I Foretell." [...] Elaida spoke again, barely moving her lips, so softly that [Rand] could barely hear her less than an arm's length away. "This, too, I Foretell. Pain and division come to the whole world, and this man stands at the heart of it. I obey the Queen," she whispered, "and speak it clearly."
-- The Eye of the World, Chapter 40: The Web Tightens
See also this question.
No, not all Foretellings are so loud and violent. For evidence, we can look at the other prominent Foreteller in the series: Elaida.
Much of Elaida's actions were based on a Foretelling at the "royal line of Andor" would be the key to winning the Last Battle; that Foretelling drives many of her decisions during the series, including why she attached herself so closely to Morgase and was so insistant that Elayne be found and returned to the Tower. It's very strongly implied that no one else knows of this Foretelling, since it's widely believed she misinterpreted it, but that possibility is never raised in the novels.
In addition, Elaida had a foretelling about Rand when she first sees him in the palace at Caemlyn, and though she does tell Morgase about it, she doesn't scream it at the top of her lungs.
The reason Gitara's Foretelling was so violent was most likely due to just how horrible and shocking the images were, and the severity of the Foretelling itself. This is also likely the reason it killed her as she gave it.
Perhaps the most amusing and definitive counterexample comes from Nicola in Chapter 14 of Lord of Chaos:
Suddenly Nicola spoke, sounding half-asleep. "The lion sword, the dedicated spear, she who sees beyond. Three on the boat, and he who is dead yet lives. The great battle done, but the world not done with battle. The land divided by the return, and the guardians balance the servants. The future teeters on the edge of a blade."
Anaiya stared at her. "What was that, child?"
Nicola blinked. “Did I say something, Aes Sedai?” she asked weakly. “I feel . . . peculiar.”
“Well, if you're going to be sick,” Anaiya said briskly, “get it over with. Linking takes some women funny the first time. We have no time to coddle your stomach.”
Anaiya was standing right next to Nicola having just formed a circle, and not only was Nicola not speaking loud enough for people to make out what she was saying, Anaiya seems to have mistook the sounds coming out of Nicola for the groans someone makes just before they vomit.
