No, they don't.
Plague javelin does damage when the person is touched by the poison cloud. However, there is another part to it. When you press A, you'll notice that there is a plague javelin AR. If you throw the javelin directly at the person, the javelin may connect, dealing physical damage. Upped or eth. Titan's do affect that damage, but not the actual poison damage. If their defense is too high, the javelin may not connect, but the poison cloud still spreads from that spot, touching them immediately.