Though I consider myself a .NET developer, I don't prefer calling it that way. c# developer sounds much better and is a much clearer message: it says that I understand both C# and .NET (because C# and .NET are tied together). I could call myself a VB.NET developer, same story there.
What is a .NET developer? I don't know, because you cannot develop with .NET, if develop is a synonym for programming. .NET is the environment, the libraries, the languages, the CLR, the CLI, JIT, the LR, the BCL, the IDE and the IL. I find it a poor job description, but it may also mean that they don't really care: either you are an F#, a C#, an IronPython or a VB.NET developer, they're all implicitly and secretly .NET developers.
What do you need? A solid understanding why ".NET" is a poor job description and ask for a more precise one. Nobody can know everything of .NET, it is simply too wide. Orientate yourself to all sides of it and do both ASP.NET and WinForms. Don't forget Silverlight, WPF etc and two or three .NET languages.
In other words: know the forest by knowing what trees and flowers it habitats and specialize in knowing a few beautiful and common ones well.