Calibre needs to be running in order to fetch your periodicals.
I have Calibre send mine to my gmail account. I then use the Kindle browser to sign in to gmail and download the newspaper.
There may another way to do this but that's the way I go.
EDIT: There is another (better way) just have Calibre email the periodical to your @kindle.com address (not the
yourname@free.kindle.com address). As long as you take receipt via wifi, you shouldn't be charged. I just tired it and no charge appeared. If you accept delivery via 3G Whispernet, you'll have to pay 15 cents per MB.