I've answered a series of related questions recently. Do a search of the site for "FLRW metric" to find relevant questions.
Anyhow, the idea of the Big Bang arose from a solution to the Einstein equations called the FLRW metric. If you assume that the universe is homogenous and isotropic (i.e. that it's basically the same everywhere) then you can solve the Einstein equation to give an equation that describes how the universe expands. This solution is called the FLRW metric, and it does seem to be a very good description of the universe as we see it today.
If you take the FLRW description of the universe and you wind time backwards, then as you approach 13.7 billion years ago the density becomes greater and greater and eventually becomes infinite. This point is known as the Big Bang. The Big Bang happens at the same time everywhere. Actually, at the moment of the Big Bang the FLRW metric predicts that the spacing between any two points falls to zero, but the universe remains infinite. This paradoxical situation is why most of think that some theory of quantum gravity will cut in before we reach the Big Bang.
So, to answer your question, the universe today seems to be well described by the FLRW metric, and the FLRW metric predicts that the Big Bang happened everywhere at the same time. To answer your specific question about the red shift, the red shift appears to be the same in all directions so this supports an isotropic universe and a single Big Bang everywhere.
It's possible that our interpretation of the experimental evidence might change as we learn more. For example you might like to have a look at this question, which discusses eternal inflation and the possibility that the Big Bang is different in different parts of the universe. At the moment there is no experimental evidence to support the idea of eternal inflation.