I think these two terms are seperate: "sterling" of course means sterling silver. and "weighted" indicates that a material has been added into an item to give it weight. This is often sand or resin, and I've see it used in candlesticks and dishes that would otherwise be top-heavy.
See this post:
http://www.925-1000.com/forum/viewtopic.php?t=4941&highlight=weightedMiles
.