The Black Riders & The Road Darkens Saga Expansions insert this in the rules section:

New Staging Rules
When playing the scenarios in The Lord of the Rings Saga
Expansions, players reveal encounter cards individually
in player order during the Staging step of the Quest phase.
Beginning with the first player, each player reveals 1
encounter card and resolves its staging before the next player
reveals a card. If an encounter card has an effect that uses the
word “you” then the encounter card is referring to the player
who revealed the card. If the revealed encounter has the
Surge keyword, the player who revealed that card reveals an
additional encounter card. Encounter cards with the
Doomed X keyword still affect each player.

Does this mean that if I reveal an encounter card, the other players can't cancel its effects through a response? Or only that I am the single target of the effects of the encounter card and, thus, only the encounter cards with the Peril Keyword would forbid the help of the other players?

If an encounter card with Peril is revealed, the way I understand it, a player can only cancel the when revealed text for himself. Then he passes the card on.

The Black Riders & The Road Darkens Saga Expansions insert this in the rules section:

New Staging Rules

When playing the scenarios in The Lord of the Rings Saga

Expansions, players reveal encounter cards individually

in player order during the Staging step of the Quest phase.

Beginning with the first player, each player reveals 1

encounter card and resolves its staging before the next player

reveals a card. If an encounter card has an effect that uses the

word “you” then the encounter card is referring to the player

who revealed the card. If the revealed encounter has the

Surge keyword, the player who revealed that card reveals an

additional encounter card. Encounter cards with the

Doomed X keyword still affect each player.

Does this mean that if I reveal an encounter card, the other players can't cancel its effects through a response? Or only that I am the single target of the effects of the encounter card and, thus, only the encounter cards with the Peril Keyword would forbid the help of the other players?

This is a good question, and I've actually mentioned it before. Unfortunately it seems we don't have an official answer.

Interpretation 1: It seems that most people play that if the encounter card does not have Peril any player may cancel it or otherwise interact with it.

Interpretation 2: Although, a strict reading of the rule book, to me, seems to imply no one could interact with it (regardless of whether or not it has Peril) since only you are revealing it (individually). Technically you cannot "table talk" about cards you've seen so if you are revealing it individually the other players have not yet seen it and could not reasonably interact with it. To me, this interpretation is bogus, though...but I'm not sure how to reconcile the rule book with my first interpretation.

If the card has been 'revealed,' everyone can see it so there's no table talk problem here (you're not 'looking' at the card, you're 'revealing' it).

Any player can use Responses when you reveal an encounter card. As you say, Gandalf196, it is only the Peril keyword provides rules that prevent other players from using their cards while you resolve the staging of your own card.