So I figured I'd get a lock set slightly better than the junky Kwikset I've got, and I bought a Schlage deadbolt and KIK set. The deadbolt will take some drilling/sanding so I'm going to save that for the weekend, but I installed the knob tonight.
When I close the door and turn the locking tab clockwise, it locks on the entry side as it should. However, I can still turn the knob from the inside and when I do it unlocks. My Kwikset wouldn't turn at all even on the inside if it was locked - that gave me the certainty that it was locked, if that makes any sense.
Is this some sort of emergency/safety mechanism so that you can get out from inside quickly if need be?