If your other big bangs are sufficiently far away and have occurred recently enough, then they would be indetectable by any means that we have used. Including COBE's beautiful pix.

Likewise if they are close enough for their radiation to have reached us, but (a different kind of) sufficiently far away, then their radiation might be detectable, but not discernable in the harsh wash of our own universe's radiation.

As far as the shape of the universe that the COBE results imply: our own universe's mass has the major and defining effect - locally. Again, I think that if you place another universe sufficiently far away, then COBE's results look pretty much the same. I am on less certain ground here. Any cosmologists in the neighborhood?