There are 3 methods to set this up
METHOD #1 : MySQL Replication
Setup MySQL Replication where the Slave has this option
Then, any DML (INSERT, UPDATE, DELETE) or DDL (ALTER TABLE) you execute will go immediately to serverB. This makes Method #1 is the fastest and most granular approach.
METHOD #2 : Copying the table to the other server
Rather than rehash, Here is an earlier post I did May 31, 2011 for this method : How do you copy a table from MySqlServer_A to MySqlServer_B?
METHOD #3 : FEDERATED Table (MyISAM Only)
Suppose mytable on serverA looks like this
CREATE TABLE mydb.mytable
You can a mapping of the target table in serverB by running this on serverA like this
CREATE TABLE mydb.mytable_remote LIKE mytable;
ALTER TABLE mydb.mytable_remote ENGINE=FEDERATED
I wrote an earlier post back on Jan 4, 2012 : fetching externally hosted db's table from within the local mysql server
Then you would have to bulk copy everything in serverA to serverB running the following on serverA:
INSERT IGNORE INTO mydb.mytable_remote SELECT * FROM mydb.mytable;