What does the SOS mechanic do?
From what I've seen, people who are playing at the same time as you are, and who are also online, and also in a dungeon, and also have an SOS sent out, can 'hear' your SOS and choose to respond to it by clicking the SOS button.
At the start of your next battle, all your characters recover 4 HP and SP for each responder, and regardless of what happens the count and the SOS call are reset once combat ends.
It's... Not exactly an overwhelming boost.