The answer to this question can be found in Chapter One of this exciting novel. The Reaping, which is the ceremony where the tributes are selected, is held in District 12 in the central square of the District, a place where people come together. Katniss describes the square in the following way:
It's too bad, really, that they hold the reaping in the square--one of the few places in District 12 that can be pleasant. The square's surrounded by shops, and on public market days, especially if there's good weather, it has a holiday feel to it. But today, despite the bright banners hanging on the buildings, there's an air of grimness.
The Reaping is therefore held in the centre of the District, and as we read the other novels in the series, we can see that each District has its own centre that is used for this purpose. What is of course so terrible about the Reaping is the way that Panem presents it as if it were something to be celebrated and a festival, when in fact it is only something that brings terrible sadness and mourning.