The word to look for is may. For example, the rules text for Bred for the Hunt says:
Whenever a creature you control with a +1/+1 counter on it deals
combat damage to a player, you may draw a card.
If a triggered ability says "may", It's up to the owner of the ability to draw a card. If he forgets, that's too bad for him.
The rules text for Sulfuric Vortex says:
At the beginning of each player's upkeep, Sulfuric Vortex deals 2
damage to that player.
It says "deals damage" instead of "you may have Sulfiric Vortex deal 2 damage". Without the "may", it's not optional. The Infraction Procedure Guide says:
You aren't allowed to miss your own triggered abilities, and your
opponent isn't required to remind you about them.
So not announcing the trigger is a rules infraction. If you do forget a trigger in a casual game (regular rule enforcement):
If a triggered ability is forgotten (or any other game error is made)
and the error is discovered later, players should alert the judge. The
judge will try and fix the error if possible and, if not, the game
will just continue.
The responsibility for announcing the trigger lies with the player who owns the ability. While other players are not required to announce the trigger, it's a good idea to do so. If the owner notices two rounds later, it will be applied retroactively.
Official tournaments are different (competitive and professional rule enforcement):
What the judge will do when you miss your triggers:
your opponent gets to decide whether he or she wants the trigger to
happen or not.
That makes for more competitive but less enjoyable games.