I'm leaning towards the, have them SEE each other, but do not interact, and branch off. Also, what I have been thinking, why not infuse the idea that Solid had with the transporting Alex and co. to 2 diff. spots? Why not send Alex to the wounded man, he heals man, he and man fight demon, while Verra and Therrisa end up somewhere totally different and they collect certain "items" essential to continuing in the game? Here is what I've been thinking for Crono's quest. It builds off of Solid's ideas.
Crono and Co.: Crono and co. (Marle, can't remember if they have their daughter or not) leave the mines/safe place and see four people, whom they ignore (Akex, man, Verra, and Therris). They enter The Forest (bom bom bom...) where they find a dead demon, who tells them of Alex and co. Crono and Marle (daughter) decide to search for Alex, to find some answers, but the demon leads them the opposite way of Alex and co. They head through the woods (Tree Spirit boss near the end) and enter a temple (Forest Temple for now). A voice tells them to find their way through the temple, and when they reach the end, a shadow approaches them and claims to be a man called Magus. The dark figure dissapears replaced by Magus, who fights you. If you win, Magus is wounded and he regains his memory, and if you lose, Marle gets wounded and Crono fights Magus one-on-one. That battle you MUST win, and we'll give Crono a crap-load of exp. Magus still regains his memory either way. But Marle is removed from your party for a while, replaced by Magus. Magus will either become a party member or he will leave you later on, we will discuss this some other time.
Thats what I got so far ^.^