Transformations can be used for a range of purposes. In particular, they are often used to ensure that assumptions of some standard parametric model are satisfied (e.g., to make residuals approximate a normal distribution).
Generally speaking, I prefer to use transformations to generate a more meaningful representation of the scale of a variable. For example, I often find it useful to do log(x+1) on count variables where x is frequency of doing something. This is because raw counts often don't reflect the scale that I wish to represent the phenomena. For example, if you have some participants who've never done something, someone else who's done it once or twice, and some participant who has done it 100 times. The log of such a variable is a more natural representation for me.
From this perspective whether transformations result in a normal distribution or not is not the main point. I see step 1 as getting the variable on a preferred metric, and step 2 as finding a model suited to representing the data. If it turns out that the data is not normally distributed, then there are a range of techniques such as bootstrapping for handling the situation.
In terms of the implication for the effect sizes, my recommendation would be to: (1) define the most meaningful metric for the dependent variable, (2) calculate effect sizes using that metric. Thus, if log transforming makes a more meaningful metric, then use that for effect size calculation purposes.
As a side point, you may find it useful to calculate effect size on both versions of the variable in order to understand the robustness of the effect to transformation.