Many cards say something to the effect of "if this check has the swashbuckling trait, add 1d4". My questions is, does it matter which "side' has the trait? You can use a pistol, which has swashbuckling, and that would give you the 1d4. But what about fighting a monster that has the swashbuckling trait? Would you still be able to add 1d4 to your combat check?
No, monsters do not add their traits to your checks against them.
More generally, any card you are making a check against doesn't add its traits to your check. For an example of where this matters, look at the "Finesse" power on Lem, Merisiel and Jirelle:
"For your check that has or is against a card that has the Finesse trait, you gain the skill Melee: Dexterity + 1"
The words in bold are needed to allow these characters to use their Melee skill to acquire weapons, because the check to acquire [defeat] doesn't take on the weapon's [monster's] traits.
If you play the card to "Determine which skill you are using" then yes, all the traits get add. Otherwise, it only adds the traits mentioned in the powers section of the card.
Determine Which Skill You’re Using. Cards that require a check specify the skill or skills you can use to attempt the check. Each check to defeat or acquire a card lists one or more skills; you may choose any of the listed skills for your check. For example, if a check lists Dexterity, Disable, Strength, and Melee, you may use any one of those skills to attempt your check. Even if your character doesn’t have any of the skills listed for a check, you can still attempt the check, but your die is a d4. Any skills you use are added as traits to the check. (For example, if your character has the skill Melee: Strength +2, and you use your Melee skill, both the Strength and the Melee trait are added to the check.)
Some cards may allow you to replace the required skill for a check with a different one. As part of this action, you may play only 1 card or use only 1 power that defines the skill you are going to use. When you play a card that does this, add that card’s traits to the check; for example, revealing the weapon Dragon Pistol for your combat check adds the Firearm, Ranged, Piercing, and Elite traits to the check.
Any card that says "For your combat check..." is an example of a card that replaces the required skill with a different one. It determines which skill you will be using for your combat check.
So, when you play a Noxious bomb for a combat check, the check has the Poison trait. And when you play Pistol +1 for your combat check, the check has the Magic trait.
But if you play the Strength spell to add to a check, the check doesn't get the Magic trait from the Strength spell.
That sounds correct, and I'll use this reasoning with my group. Do you know why cards like Scorching Ray which lists Fire as a trait also lists "with the Fire trait" in the powers box? One of our players used this as an example for why we couldn't use the traits listed on the card, saying "If they meant it to be added, they would have added it like they did for Scorching Ray."
I do know why. And see this blog post for more. Specifically, the section comparing Force Missile.