You can message a designer from search result page. When hovering over Designer's name a bubble with options appears. You can follow, message or search for icons from the designer. 

You can also find information about the designer of the icon on the icon page, as in the screenshot below.

You will need to click on the designer's name to get to their public page. You can use the Message button to write them a personal private message. 

You can keep track of your messages in the Private messages tab.

You can message an icon designer for a Custom Icon Design or for Buying icons under the Extended License.