The pet battle quests in the current MoP
build are largely incomplete and due for extensive updates in the next build, according to Cory "Mumper" Stockton
One significant change is making the tamer battles account-wide and therefore only completable on one character.
This does not apply to the dailies, so individual characters can still complete those quests once per day. [Edited because I originally misinterpreted Mumper's statement, though it is not entirely clear.]
The same character who completes the initial battle that opens up the dailies does not have to be the one to complete them. So any other character on your account can still do those quests once per day. (You know, because they're dailies. Captain Obvious needs to stop whispering in my ear.)
While Stockton says that "significant updates" will be coming in the next build, he lists highlights of what they "have been working on." This distinction may indicate that some of the updates will not necessarily make it into the next beta patch.
Mumper's full post is after the break.