This is one of those reminder-to-self style posts.
To copy mysql table tablename from database1 on server1 to database2 on server2, issue the following command (on either server)
mysqldump -h server1 -u user1 -ppassword1 database1 tablename | mysql -h server2 -u user2 -ppassword2 database2
user1, password1, user2, and password2 are the respective usernames and passwords to use on server1 and server2, respectively. Note that there is no space between the -p and password (unlike the other switches).