Does carbon dioxide from carbonated drinks enter the bloodstream?
Is there enough CO2 to matter?

A can of soda contains about 2.2 grams of carbon dioxide (see this question from Chemistry.SE).

A human outputs about 1 kg of CO2 daily from respiration, though the rate can be much higher during physical activity. So, even if all of the CO2 content of a can of soda was absorbed into your blood, that's only 0.2% of your ~resting daily CO2 output.

A normal range for [partial pressure of CO in blood] is about 38-42 mmHg.

Gas exchange

The surface area of the lungs is something like 50-75 m2.

The surface area of the stomach is much smaller, closer to 1 m2.

Therefore, gas exchange is going to be much more efficient in the lungs than at the stomach.

Result

Any CO2 is going to be fairly efficiently cleared from expiration if it gets into the blood from the stomach. It's unlikely that you could detect any appreciable change in CO2 levels beyond what happens naturally as blood travels through the circulation and picks up CO2 from the periphery, and it's even less likely this difference would be greater than the difference between rest and exercise. That isn't even accounting for the buffering capability of the blood and diffusion into other tissues (both would smooth out any rise of CO2 over time) as well as not all of the CO2 will even make it to the stomach, and the gases that make it there can also leave via the esophagus. 

As a sidenote, the Mythbusters partly addressed this issue in their episode on Mentos+coke. One of the factors they discussed is that much of the CO2 doesn't even make it to the stomach because it is released in the process of drinking, though they didn't quantify it. Therefore, CO2 is likely to have even less of an effect than described above.