Easiest way to copy a table from one database to another?

What is the best method to copy the data from a table in one database to a table in another database when the databases are under different users? I know that I can use INSERT INTO database2.table2 SELECT * from database1.table1 But here the problem is that both `database1` and `database2` are under different MySQL users. So `user1` can access `database1` only and `user2` can access `database2` only. Any idea?