Surface waters do sink when they cool as they encounter large masses of ice (as near Greenland and Antarctica), but I think it’s a pretty slow process considering the huge volume of water in the oceans. Diffusion alone without other types of mixing phenomena wouldn’t add much to the effect. Molecules in liquids tend to diffuse very slowly because they can move only a fraction of their own diameter between collisons. The random walk equation for self-diffusion is d=(2Dt)[sup]0.5[/sup] where d is the distance a typical molecule will have moved from its point of origin in time t in a liquid of self-diffusion coefficient D. For this example, let’s call d 1500 meters (about half the average depth of the oceans) and D 2.26 x 10[sup]-9[/sup] m[sup]2[/sup]/s (from my old physical chemistry textbook, but for T=25 C which is much warmer than the average temperature of the oceans). Solving for t gives t=5x10[sup]14[/sup]s or more than a million years.
Hmm using some average numbers I’ve heard, people pee 1.2 liters a day. Let’s say an average since the start of humanity of 20 years life expectancy,an average of 8760 liters of pee/human. Multipling by the number of 110 billion people in history.
There have been 9.6 * 10[sup]14 [/sup] total liters or kilo of human pee.
So it seems that even in the best[sub] worst [/sub] case(Every pee was composed of never urinated water) it is still just under 1 part in one million composed of human pee.
Of course if you add in all the mammals and reptiles etc that have ever peed it gets many times worse. And if you start to calculate what insects do as urination it might get ugly.
This could be a serious problem for the idea - there are large barriers to mixing of different bodies of water and some bodies of water can be sequestered away for some time. Water that becomes part of NADW (North Atlantic Deep Water, high density cold water formed in the Norwegian Sea and areas near Greenland) and follows the ‘Deep Ocean Conveyor’ or thermohaline circulation might not return to the surface for 1000-3000 years.
I don’t know if diffusion would be sufficient to move enough molecules out of that to get them properly mixed in the time required for you to drink it from the surface of the ocean.
On preview - I see bibliophage has addressed this a little - but 25°C seems a little warm for the oceans, especially the North Atlantic. Maybe that’s Columbus’ pee cooling off after he relieves himself at 39°C? 
Well, I was originally going to suggest that self-diffusion alone would account for thorough mixing after 500 years, but I had done my own back-of-the-envelope calculation and determined I was way off. I think that’s because I’m more of a gas dynamics guy, and the diffusion rates which are intuitive to me are a different order of magnitude.
So I backed off and said that diffusion would be of assistance in getting specific water molecules across steep density and thermal gradients which interfere with bulk transport mechanisms. I certianly don’t have the know-how, or the space on the back of this envelope, to figure out how much that would work.
Never mind.