Good idea, and repeatedly requested. Until everyone on the team can see everything, we have to lean heavily on in-mission dialog and enemy descriptions to push the story to anyone who wants it; unfortunately, unless you're fighting Captain Exposition ("Wait until you hear the details of my plan!") this isn't always easy to do.
Arc 55669 - Tales of the PPD: One Hell of a Deal (video trailer)
Arc 64511 - The Wrecking Ball
Arc 1745 - The Trouble With Trimbles
Arc 302901 - HappyCorpse
The only two things that are invisible to the entire team (at least, as much as I can recall) are the text displayed upon returning to the contact and global messages such as Boss Defeated Text.
Clicking the "!" starburst in the nav window (or the More button on the mission list) will give you the full contact dialogue for the start of the mission.
That said, I would like them to find a way to make the return dialogue visible and the global messages both more noticeable and team-wide.
Never surrender! Never give up!
Help keep Paragon City alive with the unofficial City of Heroes Tabletop Role Playing Game!
Strongly, strongly endorsed. As a story-focused Mission Architect, I want everyone on the team to be able to read what I wrote without asking the leader to repeat it.
My characters at Virtueverse
Faces of the City
Why just in the MA? I'd love the option to keep up with what's going on during Task Forces and regular arcs.
Hear, hear! Especially for the inexperienced (as in, have never done a particular TF/SF/arc before) rolling along with a group of veterans...it can be hard to get them to slow down for story. Why are we killing 8000 witches, again? 
Arc 55669 - Tales of the PPD: One Hell of a Deal (video trailer)
Arc 64511 - The Wrecking Ball
Arc 1745 - The Trouble With Trimbles
Arc 302901 - HappyCorpse
I posted this before water got sucked out of the atlantic but I'd like to post it again to hear the American (or n00bs as they are known to global historians
) pov.
In AE custom missions a lot of flavour and setup for the missions comes from contact dialogue. Foreshadowing, resolution, suprise findings etc. this is fine for solo missions because the one player gets all the info. However, on a team, only the leader gets this info, the rest just get a synopsis once the leader has accepted the mission and they also dont get the exit text.
I'd like all members of an AE team to get the full text. if they dont want to read it, they just click "ok" and get going but at least the option will be there for those that actually want to follow the story.