I have always followed Sheldon Brown's advice to measure 24 links with a ruler and if they measure 12 1/16th (i.e., they have stretched by 1/16th) then you should replace the chain if you want to avoid any damage to the rings and sprockets. If they measure 12 1/8th, the chain is seriously in need of replacement and you can assume that it has caused damage to the rings and/or sprockets.
This is easy (if you steal a ruler from your local museum) and the criteria are roughly the same - i.e., 1/16th is c0.5% of 12".
Provided you change chains when they reach 12 1/16th, you don't need to replace the sprockets every time you use a new chain, as the sprockets will outlast several chains.
However if you don't replace chains until 12 1/8th or later, you will have damaged the sprockets and you may well need to replace both at the same time.
