It doesn't work that way. The personal protection of the president is never left to others. The Chileans certainly have the responsibility for security, but they have no right to restrict Bush's Secret Service protection.
I think some Chilean SS were "unclear on the concept".
Back when Clinton was in power there were contemplations from some in the New Zealand Police that all US Secret Service agents must not carry arms when they protect Clinton if he visit NZ. "Our laws are that no weapons are to be carried by foreign bodyguards when in New Zealand. There will be no exception to whoever."
But of course things changed when Clinton eventually did make a formal visit in 1999 (for the APEC summit in Auckland) - an amendment was passed in 1998 in Parliament. I think the USSS agents were armed but not publicly mentioned so as not to upset New Zealanders. But I sensed some NZ Police members aren't happy about the arrgangement with the USSS agents - for long the NZP Police's diplomatic protection squad was responsible for visiting foreign heads of states' safety, and Australian Prime Ministers' personal bodyguards were not armed whenever an Australian PM visits here.