BuddyPress

I run a network install of WP and use BP mainly so I can use the calendar across both websites.

When a user registers, can I automatically put them in a group depending on which site they register on?

You need to create a plugin for that or hire a developer to do so.

Try the Welcome Pack plugin:

http://wordpress.org/extend/plugins/welcome-pack/

Try this plugin:

http://wordpress.org/extend/plugins/buddypress-registration-groups-1/