You can't write a partial class across projects. A partial class is a compile-time-only piece of syntactic sugar - the whole type ends up in a single assembly, i.e. one project.
All partial-type definitions meant to be parts of the same type must be defined in the same assembly and the same module (.exe or .dll file). Partial definitions cannot span multiple modules.
You cannot have two partial classes referring to the same class in two different assemblies (projects). Once the assembly is compiled, the meta-data is baked in, and your classes are no longer partial. Partial classes allows you to split the definition of the same class into two files.