The tire that "lays rubber" will be the one with the least traction (ie, the least load). In a straight line, this will be the right rear tire due to the loading of the left rear by the torque transmitted through the driveshaft.
In a turn, the inside tire will lose traction first so long as both tires are on identical surfaces.
That difference in acceleration you noticed between when both tires have traction and when one has "broken free" is due to the differences between static and kinetic friction coefficients of the tires. Once a tire loses traction, it'll offer perhaps 70% of the grip that it would have if it maintained contact with the road. To put it simply, open differentials can only transmit as much torque as can be resisted by the tire with the least amount of traction.
See the following picture:
http://i190.photobucket.com/albums/z166/barry1me/DSC01261.jpg
The carrier cross-pin (the vertical pin in the picture with spider gears on the top and bottom....it almost looks like a dumbell) will always be rotating with the ring gear along a horizontal axis as seen on your screen. So long as each tire has equal traction, the side gears (which couple to the axles and, consequentially, to your wheels) will receive equal power from the engine and will spin at the same speed. When one tire loses traction, that corresponding side gear will spin wildly out of control while the opposing side gear will barely accelerate at all.
Unless you make a habit of doing burnouts and spinning your tires in the rain, the tires will wear evenly.
