So it turns out:
The proper amount of oxygen dissolved in wort is 8-10 ppm.
Shaking typically yields around 4 ppm. It's possible to achieve as much as 8 ppm with plenty of headspace and LOTS of vigorous shaking. As an example, 5 minutes of shaking a 1.077 wort may only achieve 2.7 ppm. Siphon sprayers will be in the same range.
Air with an Oxygen Stone can also only reach 8 ppm, regardless of the amount of time the stone is in the wort.
Oxygen with an Oxygen Stone Using a .5-micron stone and a flow rate of 1 L O2 / min, you need around 60 seconds to get 9 ppm, as shown:
30 seconds pure O2 5.12 ppm
60 seconds pure O2 9.20 ppm
120 seconds pure O2 14.08 ppm
When oxygenating a higher-gravity wort, you need higher oxygen levels - roughly proportionate to the amount of yeast. However, it's usually recommended to reoxygenate after the yeast have time for a cell division, This will result in a cleaner-flavored beer. Obviously, if you want your beer to have more off-flavors (specifically acetaldehyde and diacetyl), then a second dose of oxygen is counter productive.
Looks like I need to make a trip to the pet store...
EDIT: Keep in mind these values are also temperature dependent. The cooler the wort is, the better your ppm aeration will be in the end.