Dry Brining Turkey

I suggest starting with this Cook's Illustrated brining article for more information on brining.

Much of what would be transferred into your turkey is liquid from the osmosis, but some salt, and sugar if you choose to use it, would be added as well.

If you purchase an "enhanced" turkey that has already been injected, you don't want to brine it. The injection has a similar effect. If you've made an enhanced turkey in the past, you've already eaten a bird with added sodium.

Harold McGee states that the turkey will gain 10% of its weight in a combination of water and sodium. Your brine comes out to about 3.9% salt, so your 12 lb turkey will gain ((12 lbs * .1) * .039) = .0486 lbs = 21 grams of salt from a 3.9% brine. Your average soda has 55 milligrams of salt per serving. The total amount of salt from your brine will be spread over the entire turkey, so divide that amount by the number of servings you expect that turkey to provide to get the number of additional milligrams of sodium per serving of turkey from brining.

If at the end of the day you don't want to brine, in that same article McGee also suggests a way to keep turkey moist without a brine.

EDIT: I did my math wrong above, and have corrected the formula and results.

A few things aren't quite right here.

1- a cooler full of water for 3 days.
A turkey will take 3-4 days to thaw in the fridge- between 35-40F. Quicker methods call for submerging in running water for some hours. Submerging the turkey in stagnant water- even if it started as ice water- will allow the turkey to rise well above 40F over the course of 3 days. You would have to add ice or have the turkey somewhere colder than 40F for this to be safe.

2- brining too long turning a turkey to mush.
The salt in a brine denatures some proteins and supercharges the bird with water. These will make the meat more tender and seem more tender respectively. That said- the risk isn't that the meat will get mushy. Unless you are adding some other ingredient to the brine that is actually a proteinase like papain the risk is not mushiness but being too salty.

3- 1-2 days of brining.
I have never seen a recipe that called for brining this long. The brine recipes I have seen are on the order of 4-10 hours. If your brine recipe calls for this length of time then it is probably more dilute than the ones I use. In such a case it would probably be fine to thaw it in the brine if you address the safety concern.

The turkey being frozen will prevent the brine from penetrating but as the bird thaws from the outside in the outside of the bird will have more alone time with the brine than the deeper meat. I can't say whether this would be a problem- especially as Harold McGee says that the salt from brining isn't able to penetrate very far into the meat anyway.

Unless it is cold enough outside (or you have enough fridge space) that you could brine the turkey at <40F for 2 days while it thawed I would recommend at least mostly thawing it before introducing it to your brine.