My only experience with embassies is in fiction, so bear with me if I’m completely off-base.
Movies, comic books, etc. always show the guards at embassies (US Marines, or plainclothes guards, or whoever) as armed - not the hired local security, but people of the nation the embassy is representing.
Do they get their guns purchased in-country, or delivered from the home country?
If the latter, do the guns come through the diplomatic bag, do they need to be declared or the host nation notified in any way?
Without going into details, the local US Consulate (which does many Embassy things here in the Kingdom for historical reasons) has a very small USMC detachment. Most of the guards are hired by the US Government on the Subcontinent. (I never asked them if they were Indians or Pakistanis.)
These guys make up the bulk of the guard force. When they were attacked last year, they stood tall and took the fire. Many of them died.
The guns? That is a minor problem. They almost certainly come in through the diplomatic pouch. (‘Pouch’ is a figure of speech, we are talking shipping containers here.) Do we tell the local authorities? I wouldn’t.
Oh, I forgot to mention that although the Marines guard US diplomatic countries, most other countries have their national police handle that sort of thing.
I recall reading in another thread that the diplomatic pouch/shipping container may legally be x-rayed - how does that jive with not telling the host country?
Plus, is anyone naïve enough to believe the Marines aren’t packing heat?