Via Chris Blattman, evidence that they do in economics.
Blogging about a paper causes a large increase in the number of abstract views and downloads in the same month: an average impact of an extra 70-95 abstract views in the case of Aid Watch and Blattman, 135 for Economix, 300 for Marginal Revolution, and 450-470 for Freakonomics and Krugman. [see regression table here]. These increases are massive compared to the typical abstract views and downloads these papers get- one blog post in Freakonomics is equivalent to 3 years of abstract views! However, only a minority of readers click through – we estimate 1-2% of readers of the more popular blogs click on the links to view the abstracts, and 4% on a blog like Chris Blattman that likely has a more specialized (research-focused) readership. There is some spillover of reads into the next month (not everyone reads a blog post the day it is produced), and no evidence that abstract views and downloads lead blog posts.
The paper is here. We are probably in roughly the same position as Chris Blattman in terms of readership and attention (although we cover a wider variety of papers than he does, which may cause our clickthrough rate to go down. It would be interesting to do an experiment to see if these findings bear out when one takes account of unobservables like paper quality and general interest – do some random assignment, so that e.g. a blogger only publishes posts on a randomly assigned 50% of the papers she has enough interest to write up in draft form, to see if there are any measurable differences between the published and unpublished population.