Similar to how you can choose your opponents when rematching the Champion's Cup, you're actually allowed to choose which special trainer you would like team up with to take on the Galarian Star Tournament. The first time you take on the tournament, you will only get to choose one of The Galar Region Gym Leaders or Hop as your partner. After completion, you will then get to choose from a menu. The more times you defeat the Galarian Star Tournament, the more partners you will earn on this menu. After defeating the Galarian Start Tournament with all fifteen different trainers as your partner, you will even earn Sordward and Shielbert as possible partners.
In total, there are seventeen different partners for you to earn the right to battle with on each version: Hop, Milo, Nessa, Kabu, Bea (Sword), Allister (Shield), Bede. Opal, Gordie (Sword), Melony (Shield), Marnie, Piers, Raihan, Leon, Klara (Sword), Avery (Shield), Mustard, Peony, Sordward, and Shielbert. No matter if you partner or face a trainer, their Pokemon Teams will never change.
Furthermore, some of the trainers you face will only pair up with certain other trainers. Milo will only pair with Nessa, Kabu and Piers; Nessa will only pair with Milo, Gordie, Kabu and Bea; Kabu will only pair with Milo, Nessa and Peony; Bea will only pair with Nessa, Gordie and Mustard; Bede will only pair with Opal, Klara, Hop and Shielbert; Opal will only pair with Bede, Hop, Mustard and Piers; Gordie will only pair with Bea, Nessa, Raihan and Klara; Marnie will only pair with Piers, Hop, Sordward and Klara; Piers will only pair with Marnie, Milo, Raihan and Opal; Raihan will only pair with Leon, Gordie and Piers; Leon will only pair with Raihan, Mustard, Peony and Hop; Hop will only paiur with Leon, Marnie, Bede and Opal; Mustard will only pair with Leon, Opal, Klara, Bea and Peony; Klara will only pair with Mustard, Bede, Gordie and Marnie; Avery will only pair with Mustard and Bede; Peony will only pair with Mustard, Leon and Kabu; Sordward will only pair with Marnie, Sordward; and Shielbert will only pair with Bede and Shielbert.