The design does not /have/ to involve writing a compiler. For instance, the X3J13 committee designed the ANSI Common Lisp standard, but left writing implementations up to third parties. On the other hand, the team of Julia language developers both design and write the compiler.

I guess it depends on your motivations for creating the language. Programmers will often design a language and write the compiler/interpreter themselves as a learning experience.

https://en.wikipedia.org/wiki/X3J13

http://julialang.org/