It says SHAKTI is an "open source processor" but I cannot find any design docs anywhere.

It's just RISC-V.

"RISC-V has been chosen as the instruction set architecture (ISA) for the SHAKTI class of processors. The SHAKTI processor and systems design is developed using the open source high-level synthesis (HLS) language: Bluespec System Verilog (BSV)."

Yes, but where is the actual Verilog for the cores? At least BlueSec doesn't seem to be a proprietary dialect: https://github.com/B-Lang-org/bsc (BSD License)