The read depths are technically a mixture of Poisson or log-Poisson distributions (depending on whether PCR was done), I think, but usually modelled well enough with the inverse binomial distribution. In log scale it is a bit more overdispersed than Poisson, and the second parameter in the density function for the inverse binomial distribution can be used to account for that.

It may be "normal enough" for your needs, with the understanding that there will be more "outliers" or extreme values than you'd expect to see under the normal distribution. Robust statistics (e.g. MAD or Tukey's biweight midvariance instead of standard deviation) can be used to insulate your results from these extreme values if you'd like to use statistical methods that assume normality.