I don't know much about these ... but I'm pretty sure that they must execute a Windows batch script to do the actual build, which I would guess is triggered by Hudson, etc. monitoring the directories you specify.
It's quite easy to run the Amethyst build from the command line - there are two ways to do it:
1) use VS. All you need to do is use a command something like
"C:\Program Files (x86)\Microsoft Visual Studio 9.0\Common7\IDE\devenv.exe" myproject.amproj /ProjectConfig "Debug|AnyCPU" /build
The project config just specifies the Configuration and Platform, so it could be 'Release|Android', for example. This is the easy way, but it can be a bit slow because you are activating VS for each build and of course requires a copy of VS to be installed, which you may not want to do if, say, you are running your build engine on a server.
2) Use MSBuild on the Amethyst build targets directly. This is more complicated because you have to provide MSBuild with some extra properties - like the location of the Adobe compilers, which Amethyst provides automatically. You don't need to install VS or Amethyst, but you do need to provide a copy of the AmethystBuild dll. This is faster, and probably preferred if you are integrating it into a large and serious server production run. But it is more complicated to set up. I can tell you what you need to do here if you really need to do this, but I'd stick with the simpler devenv approach initially.
Also, for faster builds use the Compiler Service (Tools | Options | Projects |Amethyst | Compiler Service). This isn't 100% yet which is why it's turned off by default. There seems to be a problem sometimes with Amethyst connecting to the compiler service when you first start it up. Other than that it's OK to use.