I know that meteor showers have a "peak" observation time.
What does the distribution in time look like? Is one side of the peak steeper/shallower or longer/shorter than the other side, and if so, why?
The meteor sightings in a meteor shower can be approximated to a normal distribution. In fact, research papers use a normal distribution to model meteor showers.
The exact characteristics of the shower will depend on the location, local weather conditions etc. The number of sightings and the number of actual meteors impacting the earth's atmospheric conditions are different as it is the local conditions that determine the number of sightings.
The International Meteor Organisation maintains data on the number of meteor sightings which shows that the sightings can be approximated to a normal distribution.
However, this data has to be used carefully because these refer to the reported sightings and the errors are significant.