Intel microcode updates on Linux are provided through regular distribution repositories. There is no phoning home feature.

Do they get updated when you run the package manager, or do they update automatically?

Binary packaging systems download microcode.dat (a text blob containing microcode) from Intel during the build process. Microcode.dat gets converted into an initramfs image that supplies the new microcode early in the system boot. Users download the built binary package, which then modifies the boot parameters of their system.

So normally only the build servers contact Intel. Source-based distributions may. But even then the source files are verified with cryptographic hashes (which are in turn signed by the maintainers' private keys).

Edit: I may have misread your question. I thought you were asking by implication if the microcode blob is pulled after the install. If not, then: No. Updates happen when the package manager decides.

https://www.kernel.org/doc/Documentation/x86/early-microcode...

https://gitlab.com/iucode-tool/iucode-tool

https://downloadcenter.intel.com/download/26400/Linux-Proces...

Either via the package manager or via the user manually downloading and loading the microcode update of their choosing. For example: http://askubuntu.com/questions/545925/how-to-update-intel-mi...