Does Sci-Compiler compile the firmware on my PC?

It can, but it does not have to: the compilation can run either on your own PC or on our servers.

Sci-Compiler generates the firmware from your block diagram, but turning that firmware into a bitstream for the FPGA always requires the tools of the FPGA vendor. Depending on which board you are targeting, that means Xilinx Vivado or Intel Quartus Prime. What you can choose is where those tools run.

On our servers. As long as your license is not expired, you can use the Remote Customization Service: the firmware is compiled on our servers, so you do not need to install, configure or license Vivado or Quartus on your computer at all.
On your PC. You install Vivado or Quartus yourself, in the exact version required by your board, and keep it configured in Sci-Compiler. A few boards also need additional Vivado licenses, such as JESD204 or 10 Gbps Ethernet.

Which of the two tools you need, in which version, and whether extra licenses are required depends on the specific product. The Supported Hardware Page lists all of this for every supported board, together with the availability of the Remote Customization Service.