To exchange data between different MySQL databases use the INSERT…SELECT syntax.
If both tables are identical:
INSERT INTO database2.table1 SELECT * FROM database1.table1 WHERE 1;
To insert the value of selected columns use:
INSERT INTO database2.table1 (column1) SELECT column1 FROM database1.table1 LIMIT 1;
This only works, if both databases are hosted on the same MySQL server! There is absolutely no way to exchange data between different/remote MySQL servers using standard (My)SQL syntax!
If you intend to access data from different/remote MySQL server, use the FEDERATED storage engine (cf. the MySQL website), which allows you to execute local queries on the remote database.