When you play From the box, so not using character decks, you can trade cards.
When you use character deck, the only way of giving card to another is to give one card to character in the same location. And after game you take those cards back.
Only one...
The upcoming Ultimate XXX can be combined somewhat with other decks. How it will be done has not been revealed fully. But you Are not allowed to mix normal character decks.
In home games it could be possible ofcourse but I supose that there would be power balance problems.
Clarification - you aren't allowed to mix cards between class/character decks in organized play. If you're just playing in a normal AP, however, you can use any cards appropriate to your deck composition and the adventure deck level. So a starting character (i.e., no scenarios played) can only have cards with the Basic trait; a character that has started adventure 3 may only have cards with the B, C, P, 1, 2 or 3 set indicators; etc.
Are you building the character for organized play (Pathfinder Society Adventure Card Guild) or regular play (e.g., Rise of the Runelords)?
If you're playing organized play and filling out chronicle sheets, then you are restricted to whatever cards are in your class deck box. You cannot trade between characters at the beginning of the scenario in organized play, however Season of Plundered Tombs lets you trade with traders while setting up. The Ultimate decks that Hannibal_pjv mentioned can help boost up some of the more problematic characters (such as Zarlova, who has basically no Arcane support in the deck for the Theurge role), but exactly how they work has not yet been revealed.