At least 7 days, but it will depend on how the private beta is going.
About an official response, this answer from Chris W. Rea has been edited by Robert Cartaino♦ in a way that now it says (emphasis mine):
I understand the typical private beta is at least 7 days, but
sometimes it gets extended if the private beta does not yet have
their scope worked out or it isn't yielding the kinds of metrics and
quality questions desired.
About the metrics, we always can check how is it going in the Area51 Proposal.
31.3 questions per day -> healthy beta
94% answered -> healthy beta
50 avid users -> we need 150 (100 more) users with 200+ rep, 10 with 2k+ rep, 5 with 3k+ rep
1.5 answer ratio -> we need more answers
273 visits/day -> 1500 needed, but that would come in open beta
But I guess the metrics are less important, if this site does not yet have their scope worked out.
In this other answer, Robert Cartaino♦ also stated (emphasis mine):
Be careful about cracking open the text books and seeding the site
with questions that been asked on every other site on the subject.
This isn't a trial, demonstration, nor a "call for questions" just to shore up your numbers. This is your actual site.
Please ask about problems you actually have.
A link in that answer to a post in the blog, Asking the First Questions, contains in my oppinion something very interesting (emphasis mine):
It's All About Design
Design doesn't just mean the obvious issues like designing the logo,
or picking colors, or coming up with a name, or writing the FAQ. The
very act of asking questions, answering questions, tagging, voting...
everything. It's all about design.
That's why early participation is really, really important. Those
earliest questions on your site say a lot about the community. So, if
you want to ask question just for the sake of asking questions, at
least make them really good ones. Ask real, expert questions.
In short, you are going to get the site you build.
Ask your first questions with an eye on the site's design. Those first questions will likely end up on the front page when potential
experts see your site for the first time. Make those first questions
exemplary questions that are worthy of imitation.
So, back to our quiz: "What is the single most important design
element of a new Q&A site?" The answer is obviously, "The questions
on the front page." Any other design issues after that are a distant