The right side shaft has a support on it, perhaps it is what is making the noise. Here's a link to look at.
http://www.realoem.com/bmw/enUS/showparts?id=RC33-USA-04-2003-R50-Mini-Cooper&diagId=31_0622
Welcome to MA!
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I believe the rule of thumb for wheel bearings is if you turn to the left and it make a bad sound it's the right wheel bearing and if you turn to right and bad sound it's the left wheel bearing. Since you replaced both sides one would think that problem was solved unless you used a lesser quality assembly and were unlucky to have the same problem reoccur. Sorry I don't anything more to add other than Welcome to Motoring Alliance !!
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Having just replaced my passenger front wheel bearing, 00Minis post is correct. When I made a left turn the sound or slight vibration would be more pronounced.
Having replaced more driveshafts than I'd care to admit if you're really feeling something through the drivetrain it's more likely the shaft than the wheel bearing. I'd replace both as it's always easier and more effective/efficient. Plus you're going to have to refill the box with gear oil - might as well only do it once. I'd also check the top motor mount. That couple of inches is crucial in having everything where it's supposed to be to work properly.
How many miles are on the car?
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The mini has 90K miles.