GRUB relies on configuration files that are typically written in the host OS -- Elementary OS or Mint, in your case. What's likely to happen is that, when you install Elementary OS, its version of GRUB will take over the boot process. It will probably make Elementary OS the default but detect the Mint installation and make it possible to boot it as a secondary OS. If you want to change your GRUB configuration, you'll need to either do so from Elementary OS or re-install GRUB from Mint.
Many people who multi-boot different Linux distributions prefer to install a boot manager that's at least somewhat independent of each distribution. On a BIOS-based system, this could be a version of GRUB that's not tied to either distribution or something else, like LILO or SYSLINUX. On an EFI-based computer, GRUB, ELILO, and SYSLINUX are all possibilities, as are rEFInd, rEFIt, and gummiboot/systemd-boot. Most of these require either manual configuration or chainloading to the distribution-centric versions of GRUB. rEFInd, though, detects kernels at boot time, so it requires little or no maintenance when each distribution updates its kernels. See my EFI Boot Loaders for Linux page for more on the EFI options.
As a broader comment, Elementary OS, Mint, and Ubuntu are all pretty similar. They've got different default GUIs, but it's possible to install multiple desktop environments in one distribution. You may be better off doing this than trying to deal with the multi-boot problems. See, among other places, this AskUbuntu question and answer for some pointers on getting started with this approach. I'm pretty sure that Mint's common desktop environments are available in Ubuntu (and therefore in its derivatives, including Elementary OS). I don't know if you can easily get an Elementary OS "look and feel" out of a stock Ubuntu or Mint, though.