>It doesn't seem to do that to me unless the last time I shut it down was on my second monitor and the next startup (of Word) the second monitor is not enabled.
>Is it possible that you have one instance running already (minimized)? In that case it will open the next offset from the other.

No, I only have one instance. Sometimes, it may remember the position for a while. Then, for several times in a row, it opens the document in various positions.