For championship play, at the local and national level I was under the understanding that they put teams in certain pools and draw them random from there. But I could be wrong
ASA Championship play Qualification (that begins with State's) has a draw (that you may attend) that care's less who you are or where you come from. I've seen 2 teams from the same org play each other round 1. (there are no pools in qualification play) - Straight Double Elim.
Now, at ASA Nationals, they will draw you in "slots" (1 seeds, 2 seeds, etc) and they will avoid placing 2 teams from the same region together in the First round only. For the 2 pool play games, you generally will play teams from other states/regions. This (the draw) is also done at a open forum that you can attend.
Any other "tournament" is generally meaningless and the host can do as he or she sees fit when putting the pools together and if you pay your fee to enter, you are giving permission to the host to place you wherever they want.