One second per second for yourself, but other observers generally disagree.

Or one could use it as an inverse of the things measured against time, so meters/second of speed can be turned into seconds flowing at a rate of length.

Does "miles per gallon" lead to similar questions about gallons passing?

> Could they pass at some rate other than one second per second?

Only, and always, from the point of view of people in different frames of reference.