Interesting question. The shamash may be re-used each night, if desired, since it's purpose is only to serve the other lights (shamash=servant). However, the shamash should also burn each night for at least a half-hour after dark, so it may not be practical to re-use it.
There are two uses of the shamash. One is to light the other candles, and the second use is to provide light so that we are not "making use" of the light of the Chanukah candles - which are to look at but not to use - but are using the light from the shamash as well. Of course, the Chanukah candle light is also there, but since the shamash is providing light it is considered as if we are using its light and not that of the Chanukah candles.
So, according to the first reason, the shamash could actually be put out after lighting and re-used. However, due to the second reason it should burn at least a half-hour after nightfall - along with the Chanukah candles - and then can either be extinguished or let burn out.
I hope this has been clear and helpful and Happy Chanuka!