Continuity mistake: While Buffy is in the hospital, they zoom in on the bedside clock changing from 2:26 to 2:27. In the next shot of Buffy, you can see the clock in the background and it reads 12:15.
Continuity mistake: When Buffy looks into the childrens' ward the sign on the door saying 'Basement Access' disappears and reappears between shots. (00:12:00 - 00:36:15)
Other mistake: In the basement when Death throws Ryan to the floor, Ryan lands pretty softly considering that he was just thrown pretty hard.
Continuity mistake: After drinking the water with the virus, Buffy looks into a room and her hair is damp and brushed back. In the next shot, it's dry and parted down the middle.
Other mistake: Buffy freaks out when she's first taken to the hospital. Joyce tells the gang it was because her cousin died when they were young, but in season one episode "Angel" Buffy has no problem being in the hospital, same with later seasons.
Chosen answer: From what we see in the series, certain girls are identifiable as potential Slayers - Kendra clearly fell into this category. The identification method is presumably mystical in nature, but the Watchers' Council are pretty effective at that sort of thing, so they're quite good at tracking down the potentials ahead of time. Not perfect, though - it does appear that Buffy herself may have slipped through the net - certainly she had no inkling of what she was until she'd already taken on the role of Slayer. It is possible, however, that this was actually cultural - an American family would hardly be likely to turn over their daughter to some strange man for 'training', so the Watchers might have chosen to keep an eye on her covertly, whereas some other cultures (like Kendra's Jamaican parents) might be more willing to believe.
Tailkinker ★