By now, we know the importance of seasoning pasta water with salt (and lots of it), but why aren't we seasoning it with peppercorns, with dried chiles, with whole and ground spices, with fresh herbs and dried herbs and bouquets garnis? Why aren't we treating our pasta water like broth?
The answer: Because it doesn't do much.
A similar question ("What can I put in my pasta water?") was asked on Reddit, and the top-voted answer is to-the-point: Go ahead and put anything you want in pasta water (within reason: you don't want to drastically change the pH level)—but the "effects will be minimal or absent." Redditor Khuervo claims that pasta just:
doesn't really hold on to anything, it doesn't really retain much water. If you compared a cooked piece of pasta to a raw piece of pasta, you would notice that the size is almost exactly the same. Rice absorbs water and increases in size because of it, this is a situation where flavouring the cooking liquid is almost the rule.
Many of you in the comments (thank you Greenstuff) pointed out that this is a bit deceiving. While Harold McGee, who Khuervo cites, explains that the outer layer of the noodle is affected in ways the center is not—"deeper within within the noodle, there's less water available, so the starch granules aren't completely disrupted: the center of the noodle therefar stays more intact than the surface"—pasta does, indeed, absorb a fair amount of water: 1.6 to 1.8 times its weight. Still, not all parts of the noodle are affected equally or at the same rate (think of al dente—still firm in the center).
And then there's question of how the size of the molecule affects its ability to penetrate the pasta. According to others on the Reddit thread, salt is better able to penetrate pasta than other flavors because it's a very tiny molecule. Plus, they argue that most of the flavor that the salt confers onto the pasta is in the water left clinging to it.
I'm no scientist—and I haven't studied the size of aroma compounds or measured the precise amount of hot liquid a pound of pasta sucks up in 8 to 10 minutes—but my own trials confirmed Khuervo's answer: I made a pot with mint, sage, crushed garlic, and red pepper flakes; a pot with 1/2 teaspoon of smoked paprika; and a pot with the water subbed out for a comparable amount of vegetable broth.
While the pasta cooked in the three souped-up liquids was discernibly different from plain pasta, the effects were subtle, and I wasn't convinced the change was in the noodles themselves: My theory was that any noticeable effects were due to fragrance-taste (smells like mint, must taste like mint) and to particulate matter (like red pepper flakes) and the barely-flavored liquid left hanging onto the linguine.
Because when you flavor the pasta cooking water, you're essentially making a very weak broth that the pasta ultimately doesn't absorb very much of. Yes gently-flavored liquid may stick the noodles' surface, but rinse them (as you would soba) and the effect is eliminated. It's still a cool party trick if you want some just-barely-detectable paprika-flavored (scented?) noodles. And it could be a fun base for a pasta salad. But to get the most flavor out of herbs, spices, lemon zest, and other ingredients you want to use to flavor the noodles, it's best to apply them to the cooked noodles themselves—rather than to dilute their power in water that you'll eventually dump out.