I have two experimental groups. Then I test their normality respectively. Result shows that one is positively skewed and the other is negatively skewed. In this case, how should I do the data transformation? Can I log transform one group and reversed-log transform the other group？
It is hard to say on this little information, except that transforming different groups differently makes no sense and that mild skewness is often acceptable. Data that are exactly normal are probably fabricated!
It is possible that no transformation will help overall, or that one transformation will help overall. It's not just the sign of the skewness that is important but also its magnitude and why it arises.
If you could post the data, or graphs of them, you would be likely to get firmer advice.