Sorry for late reply, had to finish some modding, got inspired to work by lore talk, haha.
Never questioned they had; however just that rather make it less credible to my ears that Isildur marched into what should be their territory (in their backyard, such speaking) and put up a large stone as a sign of their obligation (no matter mutual obligations, it would still be "in their face"). This perception, mind you, is that while it could been a celebrating symbol of companionship, a "statue of liberty" gifted by Isildur, I do not fancy that idea.
My more cynical view is because the demand of the sacred oath, at that special stone, do all but cry confidence and hearty union between the two rulers or peoples, haha.
To me it seems more likely he would put it just once you entered, or on the very border of, Gondor (the latter fairly reasonable if to been a 'neutral territory' for the rulers to met; see quote below), so when any Mountain men decend from their highlands down the valley the first thing they will face entering Gondor is the Black Stone, as a reminder on the deal with this realm; in the manner the Romans erected stones on their borders, to remind barbarians about that they were to enter the Empire and the consequences of hostile intent, and in a manner as with the Argonath in the later Age.
Also, if to trust one origin story (however odd a thing it seems to saved), the Stone came from Numenor [as I mention for the record], and to drop off such an artifact in the land of folk you're at so-so footing with, well just don't ring true to me.
But who knows? ^^
All I can say (and had to dig a while to find this again) is that Tolkien in one instance, while making drafts to RotK, makes it sound as a border point:
This comes from a directly rejected text obviously (JRR messed up Elendil's family tree when writing, hehe), so folks are free to deem I read it wrong or was a discarded idea, whatever strikes their fancy'[the Black Stone] was brought from Numenor, and marks still the l place where Isildur met the last king of the Dark Men of the Mountains, when he established the bounds of Gondor.'- HoME 8; XII the Last Debate; Note 26