Well Mario doesn't really have to talk. He never talks, even in games where there's no speech but just text. He just shouts and says "Mama Mia" and maybe a few other short phrases.
Sonic on the other hand, doesn't have to really talk either. He can talk just as much as Mario does with shouting and short phrases, which is preferable anyway.
Amy and Peach could both get kidnapped by the evil forces of Bowser and Eggman who have decided to work together by joining their armies. That's all the story you really need as far as I'm concerned. Mario and Sonic can have separate levels, maybe some levels where they work together to get past obstacles. Mario can bounce on Sonic's head to get over a locked door to the other side to unlock it so Sonic can get through, Sonic can get through a long course within a short amount of time to hit a switch that opens a door, etc.
Maybe some stages where you play as Tails/Luigi and Knuckles/Yoshi match-ups?
Sonic on the other hand, doesn't have to really talk either. He can talk just as much as Mario does with shouting and short phrases, which is preferable anyway.
Amy and Peach could both get kidnapped by the evil forces of Bowser and Eggman who have decided to work together by joining their armies. That's all the story you really need as far as I'm concerned. Mario and Sonic can have separate levels, maybe some levels where they work together to get past obstacles. Mario can bounce on Sonic's head to get over a locked door to the other side to unlock it so Sonic can get through, Sonic can get through a long course within a short amount of time to hit a switch that opens a door, etc.
Maybe some stages where you play as Tails/Luigi and Knuckles/Yoshi match-ups?