Note the entirety of the sneak attack rule:
With a sap (blackjack) or an unarmed strike, a rogue can make a sneak attack that deals nonlethal damage instead of lethal damage. She cannot use a weapon that deals lethal damage to deal nonlethal damage in a sneak attack, not even with the usual −4 penalty.
The order of these rules is basically what is confusing here: the only reason sneak attack needs the first line is because of the second line. It would probably be clearer if the order was reversed.
Under normal circumstances, a damage bonus is just a number that gets added on to a damage roll. Unless it specifies some damage type for the extra bonus, it just increases the damage you were dealing without affecting the type. So a bonus to damage would increase the nonlethal damage dealt when applied to a nonlethal attack.
Sneak attack, however, is special: it generally cannot be used to deal nonlethal damage. So even if you perform a nonlethal attack with a dagger, any sneak attack damage you add is still lethal. The first sentence offers an exception to this rule, however: weapons that default nonlethal damage, such as a sap or unarmed strike, deal nonlethal damage with sneak attack.
Skirmish has none of these rules, and so it falls back on the basics: it adds a bonus to your damage roll. It’s just a number. If that damage is nonlethal, it now deals more nonlethal damage. If that damage is lethal, it now deals more lethal damage. This is also true for damage types, like bludgeoning, piercing, or slashing, or even for things like fire damage, cold damage, and so on. A scout/sorcerer who moved at least 10 feet and then cast ray of frost would deal extra cold damage thanks to skirmish.
The only exceptions, per Complete Arcane, are ability damage or negative levels. Sneak attack and skirmish both simply deal negative energy damage equal to their value when “added” to an attack that deals ability damage or negative levels.