Part of the issue with ginger beer is that the stuff that makes ginger aromatic is somewhat more volatile than alcohol, and it's going to come out in the heads. If you run ginger beer (or a higher proof ginger wine) through a pot still, most of the actual ginger flavour is going to disappear completely.
Freeze distillation removes water, leaving behind the lighter fractions, and consequently leaves behind a lot more of the flavour of the original mash. Unfortunately it can leave behind undesirable flavours as well, and the stuff that causes headaches. But it is much more apt to result in an interesting product given ginger beer to begin with.
If you really want to start distilling, there are a number of homemade pot still options including the classic Kenmore water distiller. They all have various advantages and disadvantages depending on what you are trying to distill and whether you need more diffuse heating to prevent pectins from burning or whether you can live just with an immersion heating element.
--scott