Beyond what is specificially articulated in your By-Laws, who among the Board members actually executes Board "work" is up to the Board in general.
For example, the "Secretary-Treasurer" designation doesn't mean that this person is necessarily the "note taker" for all meetings. They need to monitor the finances and make sure that Board Meeting Minutes are being taken (and are ultimately responsible for this), but beyond that, the person actually executing some of these duties can be different. Duties can be separated among the Board members (or assigned by the President) as they see fit, or given to non-elected Committee Chairs. (Note, non-elected Board members or Committee Chairs don't have a vote on the Board.)