If I share one database with all slave sites will the users that sign up on slave1.domain.com be able to access domain.com, slave2.domain.com, etc? Or would they have to sign up on domain.com in order to have the permissions and access to all other sites?

The users can register in slave1.domain.com to have access to the slave2.domain.com or www.domain.com or simply domain.com.

In fact this will share the *.domain.com and you can login on any subdomains.
With Joomla 1.6, you can choose the domain part that must be considered as shared between the websites. This is a new parameter introduced by Joomla in its Global Configuration (see cookie domains).

In Joomla 1.5, this is Jms Multisite that compute the value and make the *.domain.com common between the websites subdomains.