As I have mentioned before, it doesnt have to be always a clash.
Let me categorize the following possible 'Encounters'
A. Space Chase - A dog fight/chase space mission between two rivals
B. Hunt - Planet hopping, one player hides/runs from the other
C. Clash - Both players come inside a story area, fights at the end.
D. Team Clash - Both players takes 3 companions to a story area with puzzles and stuff, competing with the other team
d.1: Beat to the finish
d.2: Clash at the end
d.3: Attack and defend
E. Cooperative - Both players come inside the story area and are forced to work together to complete the mission/defeat the boss.
F. Team Cooperative - Both players takes 2-3 companions to a story area with puzzles and stuff, cooperating with the other team
d.1: Team up to finish
d.2: Beat the boss
d.3: Attack or defend
G. Casual - Both players (and companions meet at a special event) players do the event while companions cheer or cooperate
g.1: Pod Race (Tatooine)
g.2: Parts hunt (Hoth mounting Taun tauns)
g.3: Volley(Ball casual game) (w/ companions)
There. Hope you like the ideas. I will also update this on the first page.