Dual boot Ubuntu / Ubuntu installation, non-UEFI PC. How do I delete one of the GRUBs?

First decide which of the both systems should control the grub instance in the MBR. Boot this system and run

 sudo grub-install /dev/sdX

to make sure this system now controls grub. In the second step boot the other system and run

sudo dpkg-reconfigure grub-pc

During reconfiguration of the package you will have the choice to install an instance of grub to the PBR of the system-partition or not to install an instance of grub at all.

That's all, you're done, only one OS will reinstall an instance of grub to the MBR from now on when a system-upgrade ships with a new version of grub.