Well, not to chastize - only to inform - but you've made 2 key mistakes that
have caused & complicated the matter:-(
First, Migration is a means of moving to a new system, not intended to
*duplicate* the content to it so as to get "2 for 1" (intentionally or not).
As you've found out only one install of the same ID can run at once on a
networked setup. This would have been easy enough to rectify had you not
proceded with error #2...
Never reinstall MS Office in an effort to remedy a problem - the programs
aren't usually what goes wrong or cause the issue. It is other supporting
files that need to be dealt with & those don't get touched during a
re-installation. This is further compounded by dragging stuff to the Trash.
Even a seasoned veteran doesn't know what all there is (or where it all is)
to be removed. If an uninstall does need to be done the Remove Office
utility supplied with the suite should be used - otherwise you leave bits &
pieces scattered about which interfere with any hope of progress toward a
resolution of the original issue.
Hopefully to get you back on track: see the following link for what to do to
recover any of your user information & proceed with a full Remove & Install
as further detailed there:
http://word.mvps.org/Mac/RemoveReinstall.html
Post back if you continue to have any problems.