Publish Part of Schedule?

When a week is published, all positions and shifts for all days in that week are affected.

You cannot Publish just certain days or a portion or particular group of the schedule separately.  When you Publish a week, the ENTIRE schedule for all positions is Published. 
Note, if your groups do NOT share employees, and do not need to see each other's schedules, you might consider creating a linked WhenToWork account for your different scheduling groups. This way each group can maintain and control their schedules independently, and the accounts will be linked for billing purposes (so one payment will cover the combined number of employees in all of your linked accounts). 

The pricing is the same whether you keep your employees in one account, or split them into several linked accounts. Since the employees and managers cannot see the data from another account, they would only be able to see the data within their own account.