The location of the antiquity site determines what civilization the resulting artifact is from. If you only work antiquity sites within your borders, you'll generally get artifacts from your civilization and barbarians. If your territory includes conquered cities, sites from that part of your civ could include artifacts from that civ.
Once an Archaeologist has worked an antiquity site, the civ that worked the site gets the artifact and the site disappears, regardless of the location of the site. So if an AI works an antiquity site within your borders, they get the artifact instead of you.
Working an antiquity site within the borders of another civ will trigger a diplomatic warning from the AI. You can tell them that you'll stop working sites within their borders, or ignore their warning.
I try to strike a good mix of working antiquity sites within my borders, in unclaimed territory, and also within other civ's borders (as I am able to do with open border agreements). This helps to give me as many options as possible for theming bonuses.