I would like to manage multiple sites running as part of a single wordpress network, but I don't want to personally host each of the individual sites. I don't mind if my database holds all the tables for each site, but I don't want to be the sole host for all the files, etc for each site. Is there any way to do this?
A multi-site network only has one set of files:
The database and WordPress files don't need to be on the same physical machine, but they need to have access to one another.
The WordPress files must be on one machine (hosting account).
The database can be spread across multiple database servers using sharding (See the SharDB plugin for an example).
If you want your various sites on different hosts, then you really need to have different WordPress installations. Read: Not multi-site.
There is a service available for managing multiple, standalone sites called WP Remote. You might want to look into that ...