Adler's Needle says the following:
Shootout, Boot: Choose an opposing dude at this or an adjacent location not in the shootout. That dude’s controller may have that dude join their posse. If that dude does not join, this dude gains a permanent control point. If this dude has higher grit than that dude and they do not join, that dude gets -1 influence while not at home until after Sundown.
Say that I am Player A, and am in a shootout with Player B. But this is multiplayer and I "Choose an opposing dude at this or an adjacent location not in the shootout" owned by Player C (and neither posse has previously invited dudes from Player C to join).
What happens? It would appear that Player C's dude is unable to join either posse because neither could be considered "their posse", and therefore the "if that dude does not join…" effect automatically triggers.