Couple years ago I got my letter of invitation and so on all set before leaving the States, but figured I'd tidy up odds and ends and collect my visa in Estonia. Nuh uh, bad idea. While I was meandering my way up there, the Russians got offended by something my government did elsewhere in the world and retaliated by making it more difficult to collect visas away from home. They told me I could collect my visa in a month if I waited around in Tallin. I respectfully declined.
My advice: do it all at home if you can. The visa process tends to be murky, changeable, and subject to whims of individual officials. However: mileage certainly varies.