One will find many assertions that GABA does not cross the blood-brain barrier. Some Italian research disputes this, suggesting that at high dosages it does. In addition, the impermeability of the blood-brain barrier to certain substances seems to vary between individuals.
But the blood-brain barrier issue may be moot. The hypothalamus has regions that are not shielded by the blood-brain barrier. In effect, it is the brain's dipstick into the bloodstream. It is sensitive to a vast variety of molecules that cannot otherwise access the brain. It responds to some molecules by stimulating the release of hormones (many from the pituitary, which is attached to the hypothalamus).
Research shows that GABA stimulates the release of HGH. (Once again, some have claimed that this can't be true, as GABA doesn't cross the blood-brain barrier.)
GABA also somehow promotes sleep in most people (possibly by increasing the natural secretion of melatonin. That's only a conjecture, but the hypothalamus is the relay station from the retina to the pineal gland, and is the first responder to changes in light, day length, etc.) As Rogue Nutritionist mentions, anecdotally the dose response is a bit unpredictable, with very high doses making some people less drowsy.
I use GABA sometimes, in oral doses (not sublingual) of up to 3 grams. In high doses it induces something akin to a mild case of the "niacin flush"--tingling skin on face, shoulders, and arms.
Some bodybuilders report that it leaves them short of breath--possibly because of the damping effect on the nervous system. I haven't experienced this, but it is anecdotally widespread. Combined with the flushing, this has sent a few people into a state of high anxiety, contemplating a rush to the ER. Should you choose to supplement with GABA, I'd start with low dosages and see how it affects you.
I makes me quite relaxed and sleepy, and tends to let me get a full night's deep sleep. It doesn't exactly leave me drowsy the next day...but it does leave me a bit, ummm, overrelaxed.