For a well rounded group in all four roles, and 3 different power sources (Arcane, Martial, 2 Primal):
Striker: Rogue, DEX/CHA build. Skills: Acrobatics, Bluff, Intimidate, Stealth, Streetwise, Thievery.
Leader: Bard, CHA/INT build. Skills: Arcana, Diplomacy, History, Religion, Heal.
Controller: Druid, WIS/CON build. Skills: Endurance, Heal, Insight, Nature.
Defender: Warden, STR/WIS build. Skills: Athletics, Dungeoneering, Nature, Perception.
Result: All 17 skills covered, with some overlap on Heal (with the Druid being the true person for the Heal check, the Bard picking it up for emergencies despite not having WIS primary/secondary), and Nature (shared between Druid/Warden.) Hits all four roles. And likely has a bit of fun in the process. ;)
these class skill lists are redundant with the granted skills for several reasons.
Hybrid characters. Instead of creating separate class skills lists for hybrids, they just use the same skill lists and then you just don't get the automatic training.
Multiclass feats. Often these feats give you the opportunity train one skill off the class you're MCing into's feat list. It would be rather silly if you couldn't get thievery by MCing into a rogue, but you could by Mcing into another class that didn't get it automatically.
Certain feats specifically mention class skills. There is really only one of these that doesn't fall into the category above,(Academy Prodigy from Dragon 390), but it does merit a mention.
I did some quick checking on C. Ross's idea that the class skill list will stay consistent between character variants, but that the skills you get automatically would change. I've actually found something of the opposite to be true. It seems that the variants tend to get the automatic powers, but might have a different skill list.
To specifically answer your question, no you can't double train a stat. Trained is a specific condition for a stat with specific meaning, it doesn't make sense to have something trained twice. Some classes give specific choices to start out (the Ranger gets either dungeoneering or nature), both skills are class skills so that gives the opportunity for them to choose one or both (using one of their other training spots).
Best Answer
You can always use a skill you're untrained in, and you can use it in any way not marked as (Trained Only). Being trained means higher numbers, and unlocking the trained only applications of the skill.
Consider Athletics and Acrobatics: nobody needs training to be able to jump, climb, swim or balance. (At least, adventurers are assumed to be minimally competent at this stuff.) However, Acrobatics can also be used to Reduce Falling Damage (Trained Only) — only people with special training in Acrobatics can do this.
In your specific circumstances:
Using Arcana for monster or arcana knowledge isn't trained only. The fighter can do this. It isn't unreasonable that someone would pick up a thing or two about magical stuff if they're living and adventuring in a world full of magic. To them this is just being reasonably educated about the world they live in.
Same goes for the Wizard making a Dungeoneering knowledge attempt. They can do this. They've probably picked up a thing or two, heard songs, read stories, been told fables about the Bulette (some of them actually reasonably factually accurate!), or so on.
Using Thievery to Open Lock isn't trained only. Everyone can attempt it.
In all of these cases, being untrained doesn't mean you can't do it at all. But being trained helps, because it means you're going to be better at it than anyone who isn't. If you're trained in a knowledge skill, you'll be more likely to have Expert or Master knowledge more of the time. If you're trained in Thievery, you'll pick the lock sooner than untrained people who will fail a few times before breaking the same lock, and be able to pick Paragon or Epic locks they wouldn't have a hope of opening.