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This question already has an answer here:

Does a rogue wielding two weapons get two sneak attacks or is the sneak attack damage bonus only valid for the first attack?

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marked as duplicate by KRyan pathfinder Feb 28 '18 at 15:30

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Sneak Attack Damage applies to every attack

There is no limit to how many times you can apply the sneak attack damage as long as you keep meeting the conditions for a sneak attack (emphasis mine):

The rogue's attack deals extra damage anytime her target would be denied a Dexterity bonus to AC (whether the target actually has a Dexterity bonus or not), or when the rogue flanks her target.

If there was a limit to the number of times this damage could be applied per turn it would have to be explicitly stated in the description.

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The rogue's attack deals extra damage anytime her target would be denied a Dexterity bonus to AC (whether the target actually has a Dexterity bonus or not), or when the rogue flanks her target.

“Anytime” means exactly that: you get the sneak attack for each and every attack that meets the requirements of triggering a sneak attack. There is no cooldown or frequency limit on sneak attack.

Moreover, not only can a rogue trigger sneak attack as many times as she is eligible to, she must to maintain a significant combat presence. The damage of a single sneak attack is simply too small to make the rogue much of a threat, particularly at higher levels.

Note that some ways of triggering sneak attack invalidate themselves after the first attack, however. For example, most forms of stealth leave it very difficult to continue to be hidden after the first attack (in fact, most make it hard to make even the one attack, but the rules for stealth are kind of a mess). The invisibility spell is another obvious situation where after making one attack (ending the invisibility spell), your remaining attacks may not continue to trigger sneak attack.

But in other situations, like flanking, you can and should get sneak attack as many times as possible. This fact is the very reason why rogues like two-weapon fighting so much, to get more sneak attack in.

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As long as they meet the conditions for Sneak Attack, yes

From the PFSRD,

The rogue’s attack deals extra damage anytime her target would be denied a Dexterity bonus to AC (whether the target actually has a Dexterity bonus or not), or when the rogue flanks her target.

So certainly, as long as the Rogue is flanking a target, both attacks will have the benefit of sneak attack. In my experience this is the most common way Rogues get Sneak Attacks, is via flanking.

For the case of being denied Dex bonus to AC, we look at Flat Footed.

Further in the SRD, with regards to the Flat Footed condition:

A character who has not yet acted during a combat is flat-footed, unable to react normally to the situation. A flat-footed character loses his Dexterity bonus to AC and Combat Maneuver Defense (CMD) (if any) and cannot make attacks of opportunity, unless he has the Combat Reflexes feat or Uncanny Dodge class ability.

Characters with Uncanny Dodge retain their Dexterity bonus to their AC and can make attacks of opportunity before they have acted in the first round of combat.

So until the target has a chance to act in the combat, it still counts as Flat Footed. They don't suddenly lose the Flat Footed condition once they've been attacked.

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