I interned, not at a newspaper, but for the website of a NBA franchise. Didn't last long, however, before finding my way into newspapers and getting my start as a researcher at a start-up 10K daily.
Three separate internships at two different suburban dailies in Ohio and Pennsylvania. I did one of them during the summer and it was worthless -- lots of desk work and answering phones, taking Legion baseball results. But I took a winter off from school one year and interned during basketball season. Wrote, wrote, wrote, every single day on that one, seven or eight stories a week. Basketball gamers four days a week, weekly prep basketball notes, weekly local bowling notes, Saturday wrestling tournaments. It was a ton of work, but I know my writing took a step forward during those few months.
KCBS radio in San Francisco, working for the wonderful Ted Robinson; KPIX TV in San Francisco when it was still great (Wayne Walker, Joe Fonzi, and genius producer Art Dlugach) and worked as a stringer for the Sacramento Bee, which then offered me a summer internship after I graduated. I stayed there six years.... good internship!
Though it wasn't an official internship, I worked PT on The Washington Post preps desk while in college. Officially, I was at the Pottsville (Pa.) Republican for three summers and I was hired there FT after graduation.
I interned at CKSA/CITL in Lloydminster, AB. A small radio and TV station which was both an affiliate of CBC (CKSA) and CTV (CITL). It was there that I learned my radio face wasn't going to take me far and I ran back to newsprint -- thankfully I had built up some clippings through my college paper.
Knoxville News-Sentinel Baltimore Sun Los Angeles Times Here's the kicker: I was initially turned down for all three, but in each case something came up and I got internship. Come to think of it, I sort of got my second job that way too.
Sports desk at the Greeley Tribune and a summer as a researcher at the old Rocky Mountain News (I worked there less than half a year before it closed and on some days there were so few people in such a big newsroom that I occasionally thought that everybody else had gone home)