You either need to re-save the file after fixing the references, or you should put the map file and the tilesets it is referring to in the same location relative to each other as your friend. The latter is usually the best thing to do.
So instead of going in and locating for Tiled each of these files, ask your friend where he put the map relative to the tiles and put your local file in the same relative folder.
If the tiles are referenced by an absolute path, then the map and the tiles were not on the same drive. Tell your friend to make sure to save the map file on the same drive as the tiles (and in general, to put them both within a “project folder”).