Do your muscles get tight? When are they tight? Do you ever "think" they're not tight, but then do some self massage and discover they're actually very tight? Do you know why they get tight? This isn't a scientific question, but would like to hear from people who think they're certain to their answers to these questions.
It's always difficult to believe others' experience can be radically different from our own. That's why anthropology was invented ("to make the strange familiar, and the familiar strange"). I recently talked to a bunch of old college friends about barefoot running and they were incredulous that a subset of barefoot runners have decided that stretching is bad for you. So we're not alone in marveling at those who get by without any stretching of any sort. Although, once again, the definition of what constitutes stretching can be quite narrow or broad, and there are at least four types--static, dynamic, active, and passive. Some people do yoga or sit in a squat position, for example, but don't consider this stretching. Others are so committed to an anti-stretching ideology that they use the euphemism 'lengthening.'
My lower leg muscles are almost always tight to some extent after a run, but especially so after a long, aerobic run. And yes, I don't always feel it, but when I massage them with my stick, I find that they are. So now I roll automatically, as a preventive measure. I also agree with Dutchie that the hamstrings connect to just about everything involved in running, so stretching those puppies out good is also high priority for me. I don't do any specific stretching for my upper body, just doing all the leg stuff seems to take care of that.
I may hit 30 mpw for the first time this week, so my mileage is decent (for a recreational runner), but even at lower mileage, stretching along with massaging has been key to remaining injury-free. Also keep in mind that some of the folks who don't stretch do have issues with trigger points that never seem to go away, while still others find themselves having to rest once in a while because an unexplainable pain in the metatarsals or PF flares up. No one wants to hear that stretching may help, so I've learned to shut my trap for the most part.
I also have a sit-stand desk, and it's great, and recent studies have indicated that even low-level activity like standing has great benefits. I also like to get on the floor of my office and stretch out from time to time. Helps a lot.