It works here.
I submit that any two pieces of text where one needs to be left-aligned
and the other right-aligned should probably be in two separate columns,
thus avoiding the issue altogether. However:
What did you try and what didn't work? What happened instead? Describe
your table a little bit.
And what version of OS and Word?
To left-align a word, and then right-align a different word on the same
line, you need to set a right tab at the right margin. This is the case
inside or outside a table. Tabs are confusing regardless, and more
confusing inside a table--to tab WITHIN a cell instead of to the next
cell, hit opt-tab.