I'm
wondering though, do these tablets lose effectiveness over time in any
measurable quantity?

They certainly slowly outgas chlorine, but at a rate that will take years
to amount to anything significant.

Do NOT store pool chlorinators indoors, such as in a garage. The chlorine
gas they slowly leak will cost you dearly in rapid corrosion to cars,
tools, etc.

Beware that cyanurated (so-called "stabilized") chlorine is of mixed
benefit at best because cyanurics quickly build up to levels that spoil the
sanitizing power of chlorine. See "Eliminating Cyanurics" about 2/3 down
my page:

http://www.truetex.com/poolcontrol.htm