Actually, you add the full strength bonus on primary hand attacks. From the SRD:
"Add the wielder’s Strength bonus (if any) to damage rolls for melee attacks with a light weapon if it’s used in the primary hand, or one-half the wielder’s Strength bonus if it’s used in the off hand."
"Add the wielder’s Strength bonus (if any) to damage rolls for melee attacks with a light weapon if it’s used in the primary hand, or one-half the wielder’s Strength bonus if it’s used in the off hand."



