1) If you're using shared hosting you probably don't have the privileges to drop tables, so I wouldn't worry about it too much.
2) There are several scenarios here.
2a) Working on different data
If person A and person B are working on unrelated data, or person A is issuing insert queries while person B is issuing update quer/ies (or vice versa) there is no problem; this will work flawlessly.
2b) Working on the same data
If person A and person B both update the same data the one who fires the update query last wins, since that update query will overwrite the changes from the former update query.
Does that answer your question?