Not if the person isn't a national of either country. At least currently for the US and Mexico and the US and Canada, if a person crosses the land border and is

Not if the person isn't a national of either country. At least currently for the US and Mexico and the US and Canada, if a person crosses the land border and is refused entry to the other country, there's no procedure to act as if they never left the first one; it's treated as a new entry, which may likely be refused (having been refused by another country is a risk factor for refusal). Both countries can't refuse, so one will have to detain.