Berkleeanton:
Whether or not time is part of our or the other membrane (or any other membrane for that matter), depends on two factors:
1. The dimensionality (total number of dimensions) of each brane, Ni, (i being the brane's index).
2. How many spatial dimensions each brane has, Si. Obvioulsy, Si can be either Ni or Ni-1.
In any case, what's going on in the other brane has little relevance to what's going on in ours.
The problem is this: if the higher-dimensional space(time) in which the branes are immersed doesn't have a time dimension, then there is no way to temporaly separate events outside of branes (or even inside any brane that doesn't contain its own time dimension). In that case, talking of a collision between branes is pointless, since a collision is a process, which requires time. On the other hand, if our brane doesn't contain its own time dimension, then GR's 4D description of our universe is not quite applicable. But, can you perhaps have both?
As far as I can tell, there are three possible solutions to this:
1. Our brane does not have its own time dimension, and GR is merely an excelent analogy to gravity's workings. I.e. it gives accurate results (within some limits), but provides the wrong mechanism (the wrong interpretation, if you will). This was the same situation with Newton's gravitation, except Newton's gravitation didn't provide any mechanism at all.
2. All our brane's dimensions (including time) are shared with some of the dimensions of the higher-dimensional spacetime in which it is immersed. It's like if you draw something on a piece of paper, the drawing's two dimensions are actually two of our three spatial dimensions.
3. I've heard that either M-theory or F-theory suggests that there are two temporal dimensions (don't remember which one). Perhaps one of these dimensions resides within our brane, and the other within the higher-dimensional spacetime.
So which one is it then? Can we even tell at the current stage of string-theory development?