Skill Swap's Effects stay as long as a Pokemon (on either side of the Move) does in Battle. Your Skill Swap User's Ability or the Skill Swap Target's Ability will reset, but only if they (specifically) are switched out.
Xatu (Synchronize) uses Skill Swap on Regigigas (Slow Start). Xatu switches out then back in. Xatu has Synchronize Ability again and Regigigas still has Synchronize Ability.