Hideous Laughter: On the creature’s next turn, it may attempt a new saving throw to end the effect.
Hold Person: Each round on its turn, the subject may attempt a new saving throw to end the effect.
Example Initiative:
Round 1
Mob A - Casts Hold person on PC A. PC A Fails saving throw
PC A
Mob B
1. In the example above, would PC A get a second saving throw vs Hold Person on round 1?
2. If the spell was Hideous Laughter, would PC A get a second saving throw in round 2?
If you continue reading Hideous Laughter, it says "This is a full round action that does not provoke attacks of opportunity." So, the "creature" in your example text is the subject not the caster. You can't take a full round action on an opponents turn without some fun shenanigans going on.
So they both function the same way, ie PC A gets a save on it's next turn. Just that Hideous Laughter gives a single extra save on round 2 (and not later rounds) and Hold Person is a new save every round.
I don't think my question is clear.
Hideous Laughter states it gets a saving throw on the creature's NEXT turn and Hold Person states EACH round on it's turn.
Therefore:
Round 1
Initiative 10 - Mob A Casts Hold Person on PC A and fails
Initiative 9 - PC A - Does he get a saving throw as a full round action against Hold Person?
2nd Example
Round 1
Initiative 10 - Mob A Casts Hideous Laughter on PC A and fails
Initiative 9 - PC A - no saving throw against hideous laughter
Round 2
Initiative 10 - Mob A
Initiative 9 - PC A - Gets 2nd saving throw against Hideous Laughter