I have two different databases on two different MySQL hosts. Is there a way I can set it up so that the master and slave sites use the different databases on different MySQL hosts?

I've tried it, but when I do, I get lots of error messages, such as this:

Error(1) [1142] query [CREATE OR REPLACE VIEW kos_users AS SELECT * FROM `db333623867`.`jos_users`]. DB Error: ANY command denied to user 'dbo333622596'@'%' for table 'jos_users' SQL=CREATE OR REPLACE VIEW kos_users AS SELECT * FROM `db333623867`.`jos_users`
Create User error [1142]: Query = [GRANT ALL PRIVILEGES ON `db333623867`.`jos_users` TO `dbo333622596`@`74.208.232.213` IDENTIFIED BY 'mypassword' WITH GRANT OPTION;], DB Message = [GRANT command denied to user 'dbo333623867'@'74.208.180.109' for table 'jos_users' SQL=GRANT ALL PRIVILEGES ON `db333623867`.`jos_users` TO `dbo333622596`@`74.208.232.213` IDENTIFIED BY 'mypassword' WITH GRANT OPTION;]
Error [1142] retrying query [CREATE OR REPLACE VIEW kos_users AS SELECT * FROM `db333623867`.`jos_users`]. DB Error: ANY command denied to user 'dbo333622596'@'%' for table 'jos_users' SQL=CREATE OR REPLACE VIEW kos_users AS SELECT * FROM `db333623867`.`jos_users`

It should be said that each database has a different name, on a different host, with a different user. As well, I can't change the permissions for any of the users.

The error message is clear and documented plenty of times in the forum
"command denied to user"
This mean that your MySQL user does not have enought privileges to perform the operation.

So you need to increas the MySQL user privileges and also give the access on the different DB located on the different servers.
To do that, you need the MySQL root login.
If you don't have such privileges, contact your hosting provider or the administrator of the serverS to increase your MySQL user privileges.

Does it not matter that these two databases are on two different hosts?

NO as long as you have the permission and that your PHP server and MySQL server can connect on the other MySQL server.

Remember that when you configure a joomla installation, it ask you provide a DB host name.
So you can have any host name as long as you have access on it.

If the DB are provided by the same hosting company, in general this does not cause any problem to connect on different DB host.

If you don't have access yourself to the MySQL privileges, contact your hosting company to ask him change the MySQL privileges to give you the autorisation to do it.

I've tried contacting my host to change the MySQL privileges and they won't do it. They're a great host, accept for this one part.

Is there a way to force the component to only use one username when working with one of the databases and a different username when working with the other?

NO it is not possible to provide you a work arround to bypass your MySQL user privileges.

The problem is that you are trying to share something between DB located on different host and this is only possible when modifying the MySQL user privileges.

If your hosting company reject to provide you the permission, perhaps you could try purchase a dedicated server to have the MySQL root login or change of hosting company.