The "Sweepers" are skaters from local skating clubs and they don't choose skaters at random from the audience. (Insurance restrictions alone prevent that from happening.)
You have to be a member of a skating club and/or the local Learn to Skate USA program for two reasons: 1) insurance; 2) advance notice. If you're going to an out-of-town competition, "Like" the hosting Club on Facebook and contact them if the competition's some time out. If it's in your area, join the hosting Club to be on the mailing list. (Join now - most clubs offer a "Learn to Skate USA" membership that's inexpensive.) My Club hasn't hosted Sectionals or Nationals, but the neighboring clubs that did host invited our members to participate. The rink's skating director informed the LTS group students. I think the registration forms were online (pdf) and available on paper at the rink desk.
For Nationals, they have try-outs several months before the competition. (Once, it was about 5 months prior, another time it was 8 months.)
For Sectionals, they just took anyone above a certain level with no audition. There was a fee for the dress and vest.
For Adult Nationals, the auditions took place about 2 months before the event. There was a dress code and skaters provided their own clothing.
Adult Sectionals doesn't typically have sweepers or tossies.
The age limits are ages 8-13. (Older skaters can be "Slushers," who patch the toe jump divots - only at Sectionals or Adult Nationals. At Nationals, the arena provides ice maintenance.)
One audition form stated Freeskate 1 or higher skaters only, another didn't specify a level, but they tested for one-foot balance on edges, so assume Pre-Freeskate level at minimum.
At the auditions, the skaters must demonstrate excellent skating skills (stroking, turning and stopping.) They're expecting balance on one foot, ability to swizzle properly (bent knees, control, power) and part of the audition had them skate to a toy, scoop it up and continue down the the rink. Another drill was to skate to the toy, stop completely and then pick up and skate away quickly. Speed mattered, too. Toe pushing and breaking at the waist were frowned upon. Listening skills were also vital - they can't have a skater who doesn't listen.
Be aware that hundreds of skaters try out for a handful of sweeper spots.
A lot of personality and smiles were the key to making the final rounds.