Do table officials make decisions?

Mostly no. Under World Aquatics rules (Part Six, Articles 3.5, 3.10 and 3.11):

• The secretaries' role is to record and signal events in the match and the decisions of the referees and other officials, not to make those decisions.
• The referees keep overall authority, including over possession and the shot clock, and may direct the timekeepers on timings.
• Everyone at the match, including table officials, must follow the referees' decisions and instructions.

The table does act on its own in set situations: the secretary whistles to stop play for a valid timeout, signals when an exclusion period expires, signals improper entry with the red flag (which stops play immediately), and the timekeeper signals 45 seconds and the end of a timeout and ends each period. Referees may also consult table officials if they think it helps.